Travaux sur les symétries de Lie des équations aux ... - DMA - Ens

Joël M E R K E RÉcole Normale SupérieureDépartement de Mathématiques et Applications45 rue d’Ulm, F-75230 Paris Cedex 05www.dma.ens.fr/∼merker/index.htmlmerker@dma.ens.frTravauxsur les symétries de Liedes équations aux dérivées partiellesen relation avec la géométrie CR[384 pages]Document 1, p. 2: Travail (publié) sur une caractérisation explicite des systèmesd’équations différentielles ordinaires qui sont localement équivalentsau système décrivant une particule newtonienne dans un champ de forcesnul. Calculs très difficiles d’élimination algébrico-différentielle; c’est aveccet article que j’ai appris à calculer, cf. les travaux sur Green-Griffiths.Document 3, p. 93: Travail (publié) produisant des exemples de tubes CRnon algébrisables en codimension quelconque.Document 3, p. 144: Travail (publié) donnant des bornes pour la dimensiondu groupe des symétries de Lie de certains systèmes completsd’équations aux dérivées partielles.Document 4, p. 185: Travail (récent et soumis) caractérisant explicitementles hypersurfaces analytiques réelles de C 2 qui sont localement biholomorphesà la sphère S 3 .Document 5, p. 215: Travail (récent et soumis) caractérisant l’équivalencelocale à la pseudo-sphère de Heisenberg dans C n pour n 3.Document 6, p. 233: Travail de synthèse (à paraître) contenant de trèsnombreuses idées à développer en relation avec la modernisation des travauxde Lie.[6 documents. En 2006, je me suis plus ou moins arrêté de travailler danscette direction pour étudier les œuvres originales de Lie, plus puissantes quece qui se fait actuellement dans cette direction.]

2Characterization of the Newtonianfree particle systemin m 2 dependent variablesJoël MerkerAbstract. We treat the problem of linearizability of a system of second order ordinarydifferential equations. The criterion we provide has applications to nonlinearNewtonian mechanics, especially in three-dimensional space.Let K = R or C, let x ∈ K, let m 2, let y := (y 1 ,... ,y m ) ∈ K m and lety 1 xx = F 1 (x,y,y x ) ,...... ,y m xx = F m (x,y,y x ),be a collection of m analytic second order ordinary differential equations, in generalnonlinear. We obtain a new and applicable necessary and sufficient condition inorder that this system is equivalent, under a point transformation(x,y 1 ,... ,y m ) ↦→ ( X(x,y),Y 1 (x,y),... ,Y m (x,y) ) ,to the Newtonian free particle system Y 1 XX = · · · = Y m XX = 0.Strikingly, the explicit differential system that we obtain is of first order in thecase m 2, whereas according to a classical result due to Lie, it is of second orderthe case of a single equation (m = 1).Acta Applicandæ Mathematicæ, 92 (2006), no. 2, 125–207Table of contents1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Proof of Lie’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3. Systems of second order ordinary differential equations equivalent to free particles . . . . . . . 22.4. Compatibility conditions for the second auxiliary system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .59.5. General point transformation of the free particle system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77.

§1. INTRODUCTIONSeveral physically meaningful systems of ordinary differential equationsare of second order, as for instance the free particle in m-dimensional space,the damped or undamped harmonic oscillator, coupled or not, having constanttime-dependent frequency or not, etc. Such systems are ubiquitous inNewtonian Mechanics, in Hamiltonian Dynamics and in General Relativity.Two classical major problems are to classify these systems up to point orcontact equivalence (Lie’s Grail) and to recognize when they coincide withthe Euler equations associated to a Lagrangian (inverse variational problem).In small dimensions, complete results hold (Lie, Tresse, Cartan; Darboux,Douglas). However, in arbitrary dimension, both tasks quickly exceed thehuman as well as the digital computer scale, due to the intrinsic complexityof the underlying symbolic computations (explosion, swelling) and to theexponentially increasing number of cases to be treated. We refer to Olver’smonograph [Ol1995] for a panorama of problems, methods and results.At least, as a first step in classification, with respect to applications, thereare both a mathematical and a physical interest in determining concrete, explicitand applicable (“ready-made”) criteria for a system of ordinary differentialequation to be equivalent, via a local point transformation, to a linearequation.31.1. Scalar equation. In this respect, we remind the celebrated linearizabilitycriterion for a single equation, due to Lie. Let K = R of C. Let x ∈ Kand y ∈ K. Consider a local second order ordinary differential equationy xx = F(x, y, y x ), possibly nonlinear, with a locally K-analytic right-handside 1 .Theorem 1.2. ([Lie1883], pp. 362–365; [GTW1989]; [Ol1995], p. 406) Thefollowing four conditions are equivalent:(1) y xx = F(x, y, y x ) is equivalent under a local point transformation(x, y) ↦→ (X, Y ) to the free particle equation Y XX = 0;(2) y xx = F(x, y, y x ) is equivalent to some linear equation Y XX =G 0 (X) + G 1 (X) Y + H(X) Y X ;(3) the local Lie symmetry group of y xx = F(x, y, y x ) is eightdimensional,locally isomorphic to the group PGL(3, K) of allprojective transformations of P 2 (K).1 By borrowing techniques developed in [Ma2003], this theorem as well as the next bothhold under weaker smoothness assumptions, namely with a C 2 or a W 1,∞locright-hand side.

4(4) F yxy xy xy x= 0, or equivalently F = G + y x H + (y x ) 2 L + (y x ) 3 M,where G, H, L, M are functions of (x, y) that satisfy:0 = −2G yy + 4 3 H xy − 1 3 L xx + 2(GL)y − 2G x M − 4GM x + 2 3 H L x − 4 3 H H y,0 = − 2 3 H yy + 4 3 L xy − 2M xx + 2GM y + 4G y M − 2(H M) x − 2 3 H y L + 4 3 LL x.Section 2 of this paper is devoted to a detailed exposition of the originalproof of the equivalence between (1) and (4), following [Lie1883]. In thecontemporary literature, to the author’s knowledge, there is no modern restitutionof Lie’s elegant proof, whereas the description of an alternative proofof Theorem 1.2 as a byproduct of É. Cartan’s equivalence algorithm appearsin the references [Tr1896], [Ca1924], [GTW1989], [Ol1995], [23] 2 .Lie’s Grail would comprise:• a complete classification of all Lie algebras of local vector fields; Lieachieved this task in dimension 2 over C (real case: [GKO1992]); however,as soon as the dimension is 3, the complete classification isunknown, due to the intrinsic richness of imprimitive Lie algebras ofvector fields;• a list of all the possible Lie algebras that can be realized as infinitesimalLie symmetry algebras of partial differential equations, together withtheir Levi-Malčev decomposition;• an explicit Gröbner basis of the (noncommutative) algebra of all differentialinvariants of each equation in the list.However, complete results hold only for the scalar second order ordinarydifferential equation. Some tables extracted from Lie’s GesammelteAbhandlungen and from Tresse’s prized thesis [Tr1896] may be foundin [Ol1995].1.3. Systems. Let x ∈ K, let m 2, let y := (y 1 , . . .,y m ) ∈ K m and let(1.4) y 1 xx (x) = F 1 (x, y(x), y x (x)) , . . ....,y m xx (x) = F m (x, y(x), y x (x))2 We note that in these references, the already substantial computations are stopped justafter the reduction to an {e}-structure on an eight-dimensional (local) principal bundle overthe three-dimensional first order jet space. The vanishing of two (among four) fundamentaltensors in the structure equations of the obtained {e}-structure yields two partial differentialequations satisfied by the right-hand side F(x, y, y x ), which are equivalent to (4) of Theorem1.2. We mention that with the help of Maple programming, the complete reduction toan {e}-structure on the base (not only on the principal bundle) is achieved in [HK1989],in the simpler case of so-called fiber-preserving transformations, namely point transformationsleaving invariant the “vertical” foliation {x = ct.}. To the author’s knowledge, thecomplete confirmation of Tresse’s results by means of É. Cartan’s method has never beenachieved.

e a collection of m analytic second order ordinary differential equations,possibly nonlinear, of the most general form.Motivated by physical applications and by geometrical questions, someauthors (Leach [Le1980], Grissom-Thompson–Wilkens [GTW1989],González-López [GL1988], Fels [Fe1995], Crampin-Martínez-Sarlet [CMS1996], Doubrov [Do2000], Grossman [Gr2000], Mahomed-Soh [MS2001], and others) have been interested in a least characterizingthose that have the Lie symmetry group of maximal dimension. In [Le1980],based on the belief that the equivalence between (1), (2) and (3) of Theorem1.2 would persist in the case m 2, it was conjectured that thesymmetry algebra of every linear system(1.5) y j xx = G j 0(x) +m∑my l G j 1,l (x) + ∑l=1l 1 =1y l 1x H j l 1(x),j = 1, . . .,m,of m 2 second order differential equations has a Lie symmetry group locallyisomorphic to the full transformation group PGL(m+1, K) of the projectivespace P m+1 (K). However, González-López [GL1988] infirmed thisexpectation, and produced a necessary and sufficient condition (see Corollary1.8 below) for local equivalence of such linear systems to the free particlesystem5(1.6) Y j XX= 0, j = 1, . . .,m.Applying Lie’s algorithm it is easily seen ([11]) that the free particle systemhas a local symmetry algebra of dimension ≤ m 2 +4 m+3, the bound beingattained by Y j XX= 0, j = 1, . . ., m, with group PGL(m + 1, K).In 1939, for fiber-preserving transformations only, Chern [Ch1939] conductedthe É. Cartan algorithm through absorptions of torsion, normalizationsand prolongations up to the reduction to an {e}-structure. Remarkably,in 1995, Fels [Fe1995] conducted the É. Cartan algorithm for general systemsyxx j = F j (x, y, y x ) and for general point transformations. As a byproductof the uniqueness of the obtained {e}-structure for which all invarianttensors vanish, Fels deduced in [Fe1995] that the flat system Y j XX = 0,j = 1, . . .,m, is, up to equivalence, the only system of second order possessinga symmetry group of maximal dimension. Alas, the {e}-structuresobtained by Chern and by Fels are not parametric, so that the counterpartto (4) of Theorem 1.2 was lacking as soon as m 2. The extraordinaryheaviness of É. Cartan’s method is a well known obstacle.Recently, Neut [N2003] (see also [DNP2005]) wrote a Maple program tocompute (among other {e}-structures) Fels’ tensors in a parametric, explicitway. The digital computations succeeded in the case m = 2, yielding Theorem3 of [DNP2005], a statement that may be checked to be equivalent to

6Theorem 1.7 (3) just below in the case m = 2. In the physically most meaningfulcase m = 3, a present-day computer is stuck ([N2003], [DNP2005]).Extending Lie’s less heavy computations 3 , we present here a completesolution to the characterization of Y j XX= 0 for arbitrary m 2.Theorem 1.7. Suppose m 2. The following three conditions are equivalent:(1) the system y j xx = F j (x, y, y x ), j = 1, . . .,m, is equivalent, undera local point transformation (x, y j ) ↦→ (X, Y j ) to the free particleequation Y j XX = 0;(2) the local Lie symmetry group of y j xx = F j (x, y, y x ) is (m 2 + 4 m + 3)-dimensional and locally isomorphic to PGL(m + 1, K);(3) the right hand sides F j (x, y, y x ) are of a special form, described asfollows.(i) There exist local K-analytic functions G j , H j l 1, L j l 1 ,l 2and M l1 ,l 2,where j, l 1 , l 2 = 1, . . ., m, enjoying the symmetries L j l 1 ,l 2= L j l 2 ,l 1and M l1 ,l 2= M l2 ,l 1and depending only on (x, y) such thatF j (x, y, y x ) may be written as the following specific cubic polynomialwith respect to y x :y j xx = G j +m∑l 1 =1y l 1x H j l 1+m∑m∑l 1 =1 l 2 =1y l 1x y l 2x L j l 1 ,l 2+ y j x ·m∑m∑l 1 =1 l 2 =1y l 1x y l 2x M l1 ,l 2.(I)(ii) The functions G j , H j l 1, L j l 1 ,l 2and M l1 ,l 2satisfy the followingexplicit system of four families of first order partial differentialequations:⎧⎪⎨⎪⎩0 = − 2 G j + 2 y l 1 δj l 1G l 2+ y l 2 Hj l 1 ,x − δj l 1H l 2m∑+ 2 G k L j l 1 ,k − 2 ∑ m δj l 1G k L l 2l2 ,k +k=1+ 1 2 δj l 1m∑k=1H k l 2H l 2k− 1 2k=1m∑Hl k 1H j k ,k=1l2 ,x +3 Throughout the article, we do not adopt the summation convention, because in severalsubsequent equations, some repeated indices shall appear that will not be summed. Also,we always put commas between the indices. For instance L j l 1,l 3,y l 2 denotes ∂Lj l 1,l 3/∂y l2shortly. As usual, δ j i is the Kronecker symbol.

7where the indices j, l 1 vary in {1, 2, . . ., m};⎧0 = − 1 2 Hj + 1 l 1 ,y l 26 δj l 1H l 2+ 1l 2 ,y l 23 δj l 2H l 1+ l 1 ,y l 1+ L j l 1 ,l 2 ,x − 1 3 δj l 1L l 2l2 ,l 2 ,x − 2 3 δj l 2L l 1l1 ,l 1 ,x +(II)⎪⎨⎪⎩+ G j M l1 ,l 2− 1 3 δj l 1G l 2M l2 ,l 2− 2 3 δj l 2G l 1M l1 ,l 1++ 1 ∑ m3 δj l 1G k M l2 ,k − 1 3 δj l 2− 1 2k=1m∑H j k Lk l 1 ,l 2+ 1 2k=1(+ δ j 1l 16(+ δ j 1l 23m∑k=1m∑k=1m∑k=1m∑Hl k 1L j l 2 ,k +k=1H l 2kL k l 2 ,l 2− 1 6H l 1kL k l 1 ,l 1− 1 3m∑k=1m∑k=1G k M l1 ,k−H k l 2L l 2l2 ,kH k l 1L l 1l1 ,kwhere the indices j, l 1 , l 2 vary in {1, 2, . . ., m};⎧0 = L j − l 1 ,l 2 ,y l 3 Lj + l 1 ,l 3 ,y l 2 δj l 3M l1 ,l 2 ,x − δ j l 2M l1 ,l 3 ,x+))+,(III)(IV)⎪⎨⎪⎩+ 1 2 Hj l 3M l1 ,l 2− 1 2 Hj l 2M l1 ,l 3++ 1 ∑ m2 δj l 1Hl k 3M l2 ,k − 1 2 δj l 1k=1+ 1 ∑ m2 δj l 3Hl k 1M l2 ,k − 1 2 δj l 2+k=1m∑k=1∑ mk=1m∑mL k l 1 ,l 3L j l 2 ,k − ∑L k l 1 ,l 2L j l 3 ,k ,k=1k=1H k l 2M l3 ,k+H k l 1M l3 ,k+where the indices j, l 1 , l 2 , l 3 vary in {1, . . .m}; and{m∑m∑0 = M l1 ,l 2 ,y l 3 − M l1 ,l 3 ,y l 2 − L k l 1 ,l 2M l3 ,k + L k l 1 ,l 3M l2 ,k,k=1where the indices l 1 , l 2 , l 3 vary in {1, . . ., m}.Let us provide commentaries and explanations. The form of the righthandside of (3)(i) is the analog of the form of the right-hand side F in (4)of Lie’s Theorem 1.2. However, we notice that the right-hand side of (3)(i)k=1

8is not the most general degree three polynomial in the variables y j x, j =1, . . ., m: some coefficients of the cubic terms vanish.Very strikingly, the differential system (I), (II), (III), (IV) satisfied by thefunctions G j , H j l 1, L j l 1 ,l 2, M l1 ,l 2is of first order for m 2, whereas thesystem (4) satisfied by G, H, L, M in Lie’s Theorem 1.2 is of second orderfor m = 1. This confirms the main theorem of González-López [GL1988],that is recovered here in the special linear case (1.5).Corollary 1.8. ([GL1988]) For m 2, a linear (nonhomogeneous) systemyxx j = G j 0(x)+ ∑ ml=1 yl G j 1,l (x)+∑ ml 1 =1 yl 1 x H j l 1(x) is equivalent to Y j XX = 0if and only if there exists a function B(x) such that the m × m matrix G 1may be written under the specific form:(1.9) G j 1,l = 1 2 Hj l− 1 m∑H j k4Hk l + δ j l B.k=1We also mention that Fels obtained a characterization of equivalence tothe free particle Y j XX= 0, j = 1, . . ., m, by the vanishing of two (nonexplicit)tensors ˜S j jikland ˜P i (Corollary 5.1 in [Fe1995]). In this reference,some parametric computations are achieved after restricting the initial G-structure, together with its subsequent prolongations, to the identity elementof the group; explicit expressions of ˜S j ∣ikl Idand of ˜P j ∣i Idare then obtained,through already hard computations. The vanishing of the two tensors ˜P jiand ˜S j iklat the identity of the structure group, namely, as computed inLemma 4.1 of [Fe1995], yields (translating into our notation)(1.10) ⎧⎪⎨⎪⎩0 = (˜S j ikl ) ∣∣∣Id= F j y i xy k xy l x− 1n + 2m∑l 1 =10 = ( ˜P∣ ji ) ∣∣Id= 1 ) (F2 D j y− F j xi y− 1 i 4− 1 m δj i[12 D ( m∑k=1∑δ j σ(l) F l 1y l 1σ∈S 3m∑F k y k xk=1)F j y k xF k y i x −−x yxσ(i) yxσ(k)m∑F k y− 1 k 4k=1,m∑k=1]m∑F l y F k x k y,xlwhere D is the total differentiation operator ∂ +∑ m∂x l=1 yl x ∂ + ∑ m∂y l l=1 F l ∂ ,∂yxland where i, j, k, l = 1, . . .,m. Strikingly, one may check that the firstequation is equivalent to (3)(i) of Theorem 1.2 and then that the secondequation yields the (complicated) four families of first order partial differentialequations (I), (II), (III) and (IV). So, in Corollary 5.1 of [Fe1995], onemay replace the vanishing of ˜S j jikland of ˜P i by the vanishing of ˜S j ∣ikl Idandof ˜P j ∣i Id, which were explicitly computed there.l=1

This phenomenon could be explained as follows: as soon as the tensors˜S j iklvanish, the system enjoys a projective connection (appendixof [Fe1995]); with such a connection, the tensors ˜P ji then transform accordingto a specific rule via tensorial rotation formulas and their general expressionmay be deduced from their expression at the identity 4 . We have checkedthis, but as we try to avoid the method of equivalence, details will not be reproducedhere. Similar observations appear in Hachtroudi [Ha1937].Even if the expressions (1.10) are more compact than the (equivalent)conditions in Theorem 1.7 (3), we prefer the complete expressions of Theorem1.7 (3), since they are more explicit and ready-made for checkingwhether a physically given nonlinear system is equivalent to a free particle.If the reader prefers compact expressions and “short” theorems, (s)hemay replace the conditions of Theorem 1.7 (3) by (1.10).Open problem 1.11. Characterize explicitly the linearizability of a Newtoniansystem in m 2 degrees of freedom, i.e. local equivalence to :m∑m(1.12) Y j XX = Gj 0(X) + Y l G j 1,l (X) + ∑Y l 1X Hj l 1(X).l=11.13. Organization, avertissement and acknowledgment. Section 2 is devotedto a thorough restitution of Lie’s original proof of the equivalence between(1) and (4) in Theorem 1.2. Section 3 is devoted to the formulationof combinatorial formulas yielding the general form of a system equivalentto Y j XX= 0, j = 1, . . ., m, under a local K-analytic point transformation(x, y j ) ↦→ (X, Y j ), for general m 2 ; the proof of the main technicalLemma 3.32 is exposed in Section 5. Section 4 is devoted to the final proofof the equivalence between (1) and (3) in Theorem 1.7, the equivalence between(1) and (2) being already proved by Fels [Fe1995].Some word about style and intentions. We wanted the proof of the equivalencebetween (1) and (3) be totally complete, every tiny detail being rigorouslyand patiently checked. This is why we decided to carefully detaileach intermediate computational step, seeking first the combinatorics of theformal calculations in the case m = 1 and devising then the underlying combinatoricsfor the case m 2. Actually, the size of differential expressionsis relatively impressive, as will become soon evident. Thus, no intermediatesymbolic computation will be hidden, hence essentially no checking workis left to the reader, as would have been the case if we did not have typed allthe computations.4 Similar rotation formulas are known in the much simpler case of (pseudo-) Riemannianmetrics, see Chapter 12 of Olver [Ol1995]. It would be interesting to write a program,finer and more efficient than [N2003], which would systematically recognize such rotationformulas in any application of É. Cartan’s equivalence method.l 1 =19

10We also would like to point out that except in every specific standardsituations, as with the much studied Riemann and Ricci tensors, presentdaycomputer programs are not yet powerful enough to apply the É. Cartanequivalence method when the number of some collection of variables is ageneral integer. All the formulas obtained in Sections 2, 3 and 4 were firsttreated completely by hand and then, some of them were confirmed afterwardswith the help of MAPLE release 6 in the cases m = 2 and m = 3.The author is indebted to Sylvain Neut and to Michel Petitot, from the Universityof Lille 1, for their help in computer machine confirmations.§2. PROOF OF LIE’S THEOREM2.1. Argument. This preliminary section contains a detailed exposition ofLie’s original proof of the equivalence between (1) and (4) in Theorem 1.2.Since our goal is to guess the combinatorics of computations in several variables,it will be a crucial point for us to explain thoroughly and patientlyeach step of Lie’s computation. Without such an intuitive control, it wouldbe hopeless to conduct any generalization to several variables. Hence weshall respect a fundamental principle: always explain clearly and completelywhat sort of computation is achieved at each step. Also, we shall many timesintroduce some appropriate new notation.2.2. Combinatorics of the second order prolongation of a point transformation.Let K = R or C. Let (x, y) ↦→ (X(x, y), Y (x, y)) be a localK-analytic invertible transformation, defined in a neighborhood of the originin K 2 , which maps the second order differential equation y xx = F(x, y, y x )to the flat equation Y XX = 0. By assumption, the Jacobian determinant(2.3) ∆(x|y) :=∣ X ∣x X y ∣∣∣Y yis nowhere vanishing. Since the equation Y XX = 0 is left unchanged by anyaffine transformation in the (X, Y ) space, we can (and we shall) assume thatthe transformation is tangent to the identity at the origin, namely the aboveJacobian matrix equals the identity matrix at (x, y) = (0, 0).The computation how the differential equation in the (X, Y ) coordinatesis related to the differential equation in the (x, y)-coordinates is classical,cf. [Lie1883], [Tr1896], [BK1989], [Ib1992]: let us remind it. A local graph{y = y(x)} being transformed to a local graph {Y = Y (X)}, we have adirect formula for the first derivative Y X :(2.4) Y X := dYdXY x=dx · ∂Y (x, y(x))/∂xdx · ∂X(x, y(x))/∂x = Y x + y x Y yX x + y x X y.

This yields the prolongation of the transformation to the first order jet space.For the second order prolongation, introducing the second order total differentiationoperator (which geometrically corresponds to differentiation alonggraphs {(x, y(x))}) defined by(2.5) D := ∂∂x + y x∂∂y + y xx∂∂y x,we may compute, simplify and reorder the expression of the second orderderivative in the (X, Y )-coordinates:(2.6) ⎧Y XX := d2 YdX ≡ DY X2 DX = D [(Y x + y x Y y )(X x + y x X x ) −1 ]=X x + y x X y⎪⎨⎪⎩1=[X x + y x X y ] {y 3 xx [X x Y y − Y x X y ] + X x Y xx − Y x X xx ++ y x [2(X x Y xy − Y x X xy ) − (X xx Y y − Y xx X y )] +11+ y x y x [X x Y yy − Y x X yy − 2(X xy Y y − Y xy X y )]++ y x y x y x [−(X yy Y y − Y yy X y )]} .Even if not too complicated, the internal combinatorics of this expressionhas to be analyzed and expressed thoroughly. First of all, as Y XX = 0 byassumption, we may erase the cubic factor [X x +y x X y ] −3 . Next, as the factorof y xx in the right-hand side of (2.6), we just recognize the Jacobian ∆(x|y)expressed in (2.3) above. Also, all the other factors are modifications of theJacobian ∆(x|y), whose combinatorics may be understood as follows.There exist exactly three possible distinct second order derivatives: xx, xyand yy. There are also exactly two columns in (2.3). By replacing each ofthe two columns of first order derivative in ∆(x|y) by any column of secondorder derivative (leaving X and Y unchanged), we may build exactly sixnew determinants(2.7){ ∆(xx|y) ∆(xy|y) ∆(yy|y)∆(x|xx) ∆(x|xy) ∆(x|yy)where for instance{(2.8) ∆(xx|y) :=∣ X ∣ xx X y ∣∣∣ and ∆(x|xy) :=Y xx Y y∣X xX xyY x Y xy∣ ∣∣∣.Hence, by rewriting (2.6), we see that the equation y xx = F(x, y, y x ) equivalentto Y XX = 0 may be written under the general explicit form, involving

12determinants(2.9)⎧⎪⎨ 0 = y xx ·∣ X ∣ x X y ∣∣∣ +Y x Y y∣ X ∣ {x X xx ∣∣∣ + yY x Y x · 2xx∣ X ∣ x X xy ∣∣∣ −Y x Y xy∣ X ∣}xx X y ∣∣∣+Y xx Y y{∣ ∣ ∣∣∣⎪⎩X+ y x y x · x X yy ∣∣∣ − 2Y x Y yy∣ X ∣}{xy X y ∣∣∣ + yY xy Y x y x y x · −y∣ X ∣}yy X y ∣∣∣Y yy Y yor equivalently, after solving in y xx , i.e.∆(x|y):(2.10)⎧⎪⎨ y xx = − ∆(x|xx)∆(x|y) + y x ·⎪⎩+ (y x ) 2 ·{−2 ∆(x|xy)∆(x|y) + ∆(xx|y)∆(x|y)}{− ∆(x|yy)∆(x|y) + 2∆(xy|y) ∆(x|y)after dividing by the Jacobian+ (y x ) 3 ·}+{ ∆(yy|y)∆(x|y)At this point, it will be convenient to slightly contract the notation by introducinga new family of square functions as follows. We first index thecoordinates (x, y) as (y 0 , y 1 ), namely we introduce the two notational equivalences(2.11) y 0 ≡ x, y 1 ≡ y ,which will be very convenient in the sequel, especially to write down generalcombinatorial formulas anticipating our treatment of the case of m 2 dependentvariables (y 1 , . . .,y m ), to be achieved in Sections 3, 4 and 5 below.With this convention at hand, our six square functions □ k 1y j 1y , symmetricj 2with respect to the lower indices, where 0 j 1 , j 2 , k 1 1, are defined by(2.12)⎧⎪⎨⎪⎩□ 0 xx := ∆(xx|y)∆(x|y) ,□ 1 xx:=∆(x|xx)∆(x|y) ,□0 xy := ∆(xy|y)∆(x|y) ,□1 xy:=∆(x|xy)∆(x|y) ,□0 yy := ∆(yy|y)∆(x|y) ,□1 yy:=∆(x|yy)∆(x|y) .Here of course, the upper index designates the column upon which the secondorder derivative appears, itself being encoded by the two lower indices.Even if this is hidden in the notation, we shall remember that the squarefunctions are explicit rational expressions in terms of the second order jetof the transformation (x, y) ↦→ (X(x, y), Y (x, y)). However, we shall beaware of not confusing the index in the square functions with a second orderpartial derivative of some function “□ j ”, denoted by the square symbol:indeed, the partial derivatives are hidden in some determinant.At this point, we may summarize what we have established so far.Lemma 2.13. The equation y xx = F(x, y, y x ) is equivalent to the flat equationY XX = 0 if and only if there exist two local K-analytic functions}.

X(x, y) and Y (x, y) such that it may be written under the form(2.14)y xx = −□ 1 xx +y x ·(−2□ 1 xy + □0 xx)+(yx ) 2 ·(−□ 1 yy + 2 □0 xy)+(yx ) 3 ·□ 0 yy .At this point, for heuristic reasons, it may be useful to compare the righthandside of (2.14) with the classical expression of the prolongation to thesecond order jet space of a general vector field of the form L := X(x, y) ∂ + ∂xY (x, y) ∂ , which is given, according to [Lie1883], [Ol1986], [BK1989], by∂y(2.15) ⎧L (2) = X ∂⎪⎨ ∂x + Y ∂ ∂y + [ Y x + y x · (Y y − X x ) + (y x ) 2 · (−X y ) ] ∂+∂y x+ [ Y xx + y x · (2Y xy − X xx ) + (y x ) 2 · (Y yy − 2X xy ) + (y x ) 3 · (−X yy )+⎪⎩+ y xx · (Y y − 2 X x ) + y x y xx · (−3 X y )]∂∂y xx.We immediately see that (up to an overall minus sign) the right-hand sideof (2.14) is formally analogous to the second line of (2.15) : the letter Xcorresponds to the symbol □ 0 and the letter Y corresponds to the symbol□ 1 . This analogy is no mystery, just because the formula for L (2) is classicallyobtained by differentiating at ε = 0 the second order prolongation[exp(εL)(·)] (2) of the flow of L !In fact, as we assumed that the transformation (x, y) ↦→(X(x, y), Y (x, y)) is tangent to the identity at the origin, we maythink that X x∼ = 1, Xy∼ = 0, Yx∼ = 0 and Yy∼ = 1, whence the Jacobian∆(x|y) ∼ = 1 and moreover13(2.16){□0xx∼ = Xxx , □ 0 xy ∼ = X xy , □ 0 yy ∼ = X yy ,□ 1 xx ∼ = Y xx , □ 1 xy ∼ = Y xy , □ 1 yy ∼ = Y yy .By means of this (abusive) notational correspondence, we see that, up to anoverall minus sign, the right-hand side of (2.14) transforms precisely to thesecond line of (2.15). This analogy will be useful in devising combinatorialformulas for the generalization of Lemma 2.13 to the case of m 2variables (y 1 , . . .,y m ), see Lemmas 3.22 and 3.32 below.2.17. Continuation. Clearly, since the right-hand side of (2.14) is a polynomialof degree three in y x , the first condition of Theorem 1.2 (4) immediatelyholds. We are therefore led to establish that the second condition isnecessary and sufficient in order that there exist two local K-analytic functionsX(x, y) and Y (x, y) which solve the following system of nonlinearsecond order partial differential equations (remind that the second order jet

14of (X, Y ) is hidden in the square functions):⎧G = − □ 1 xx⎪⎨,H = − 2 □ 1 xy(2.18)+ □0 xx ,L = − □ 1 yy + 2 □ 0 xy,⎪⎩M = □ 0 yy .In the remainder of this section, following [Lie1883], p. 364, we shall studythis second order system by introducing two auxiliary systems of partialdifferential equations which are complete, and we shall see in §2.38 belowthat the compatibility conditions (insuring involutivity, hence complete integrability)of the second auxiliary system exactly provide the two partialdifferential equations appearing in Theorem 1.2 (4).2.19. First auxiliary system. We notice that in (2.18), there are two moresquare functions □ 0 xx , □0 xy , □0 yy , □1 xx , □1 xy , □1 yy , than functions G, H, L andM. Hence, as a trick, let us introduce six new independent functions Π k 1j 1 ,j 2of (x, y), symmetric with respect to the lower indices, for 0 j 1 , j 2 , k 1 1and let us seek necessary and sufficient conditions in order that there existsolutions (X, Y ) to the first auxiliary system:{□0xx = Π 0 0,0 , □0 xy = Π0 0,1 , □0 yy = Π0 1,1 ,(2.20)□ 1 xx = Π1 0,0 , □1 xy = Π1 0,1 , □1 yy = Π1 1,1 .According to the (aprooximate) identities (2.16), this system looks like acomplete second order system of partial differential equations in two variables(x, y) and in two unknowns (X, Y ). More rigorously, by means ofelementary algebraic operations, taking account of the fact that X x∼ = 1,X y∼ = 0, Yx∼ = 0 and Yy∼ = 1, one may transform this sytem in a true secondorder complete system, solved with respect to the top order derivatives,namely of the form{Xxx = Λ 0 0,0 , X xy = Λ 0 0,1 , X yy = Λ 0 1,1 ,(2.21)Y xx = Λ 1 0,0 , Y xy = Λ 1 0,1 , Y yy = Λ 1 1,1 ,where the Λ k 1j 1 ,j 2are local K-analytic functions of (x, y, X, Y, X x , X y , Y x , Y y ).For such a system, the compatibility conditions [which are necessary andsufficient for the existence of a solution (X, Y )] are easily formulated:{(Λ00,0 ) y = (Λ 0 0,1) x , (Λ 0 0,1) y = (Λ 0 1,1) x ,(2.22)(Λ 1 0,0 ) y = (Λ 1 0,1 ) x, (Λ 1 0,1 ) y = (Λ 1 1,1 ) x.Equivalently, we may express the compatibility conditions directly with thesystem (2.20), without transforming it to the form (2.21). This direct strategywill be more appropriate.

2.23. Compatibility conditions for the first auxiliary system. Indeed, tobegin with, let us remind that the ∆(·|·) are determinant, hence we have theskew-symmetry relation ∆(x a y b |x c y d ) = −∆(x c y d |x a y b ) and the followingtwo formulas for partial differentiation{ [∆(x a y b |x c y d ) ] = x ∆(xa+1 y b |x c y d ) + ∆(x a y b |x c+1 y d ),(2.24)[∆(x a y b |x c y d ) ] y = ∆(xa y b+1 |x c y d ) + ∆(x a y b |x c y d+1 ).With these formal rules at hand, as an exercise, let us compute for instancethe following cross differentiation (remember that the lower index in thesquare functions is not a partial derivative):(2.25) ⎧⎪⎨⎪⎩(□0xx)==)( ) ∆(xx|y)− ∂∆(x|y) ∂x( ) ∆(xy|y)=∆(x|y)−( □ 0 y xy = ∂ x∂y1{∆(xxy|y) · ∆(x|y)[∆(x|y)] 2 a+ ∆(xx|yy) · ∆(x|y)−−∆(xy|y) · ∆(xx|y) b− ∆(x|yy) · ∆(xx|y)−−∆(xxy|y) · ∆(x|y) a− ∆(xy|xy) · ∆(x|y) c+}+∆(xy|y) · ∆(xx|y) b+ ∆(xy|y) · ∆(x|xy) =1{∆(xx|yy) · ∆(x|y) − ∆(x|yy) · ∆(xx|y)+[∆(x|y)]2+∆(xy|y) · ∆(x|xy)}.Crucially, we observe that the third order derivatives kill each other anddisappear, see the underlined terms with a appended. Also, two productsof two determinants ∆(·|·) involving a second order derivative upon onecolumn of each determinant kill each other: they are underlined with bappended. Finally, by antisymmetry of determinants, the term ∆(xy|xy) ·∆(x|y) vanishes gratuitously: it is underlined with c appended.However, there still remains one term involving second order derivativesupon the two columns of a determinant: it is ∆(xx|yy).We must transform this unpleasant term ∆(xx|yy) · ∆(x|y) and expressit as a product of two determinants, each involving a second order derivativeonly in one column. To this aim, we have:Lemma 2.26. The following three relations between the differential determinants∆(·|·) hold true:(2.27) ⎧⎪⎨∆(xx|xy) · ∆(x|y) = ∆(xx|y) · ∆(x|xy) − ∆(xy|y) · ∆(x|xx),∆(xx|yy) · ∆(x|y) = ∆(xx|y) · ∆(x|yy) − ∆(yy|y) · ∆(x|xx),⎪⎩∆(xy|yy) · ∆(x|y) = ∆(xy|y) · ∆(x|yy) − ∆(yy|y) · ∆(x|xy).15

16Proof. Each of these three formal identities is an immediate direct consequenceof the following Plücker type identity, easily verified by developingall the determinants :(2.28)∣ A ∣ 1 B 1∣∣∣·B 2∣ C ∣ 1 D 1 ∣∣∣ =C 2 D 2∣ A ∣ 1 D 1∣∣∣·D 2∣ C ∣ 1 B 1 ∣∣∣ −B 2∣ B ∣ 1 D 1∣∣∣·D 2∣ C ∣1 A 1 ∣∣∣,C 2 A 2A 2A 2where the variables A 1 , A 2 , B 1 , B 2 , C 1 , C 2 , D 1 , D 2 ∈ K are arbitrary.Thanks to the second identity (2.27), we may therefore transform the resultleft above in the last two lines of (2.25); as desired, it will remain determinantshaving only one second order derivative per column, so that afterdivision by [∆(x|y)] 2 , we discover a quadratic expression involving only thesquare functions themselves:(2.29) ⎧( )□0 − ( xx □ 0 y xy)x = 1{∆(xx|y) · ∆(x|yy) − ∆(yy|y) · ∆(x|xx)−[∆(x|y)]2⎪⎨−∆(x|yy) · ∆(xx|y) + ∆(xy|y) · ∆(x|xy)}1= {−∆(yy|y) · ∆(x|xx) + ∆(xy|y) · ∆(x|xy)}[∆(x|y)]2⎪⎩= −□ 0 yy · □1 xx + □0 xy · □1 xy .In sum, the result of the cross differentiation (□ 0 xx) y − (□ 0 xy) x is a quadraticexpression in terms of the square functions themselves! Following the samerecipe (with no surprise), one may establish the following relations, listingall the compatibility conditions (the first one is nothing else than (2.29)):(2.30) ⎧( )□0 − ( )xx □ 0 y xy = − x □1 xx · □0 yy + □1 xy · □0 xy( ),⎪⎨ □0 − ( )xy □ 0 y yy = − x □0 xy · □0 yx − □1 xy · □0 yy + □0 yy · □0 xx + □1 yy · □0 xy( ),□1 − ( )xx □ 1 y xy = − x □0 xx · □1 yx − □1 xx · □1 yy + □0 xy · □1 xx + □1 xy · □1 xy ,⎪⎩( )□1 − ( )xy □ 1 y yy = − x □0 xy · □1 yx + □0 yy · □1 xx .Instead of checking patiently each of the remaining three cross above crossdifferentiation identities, it is better to establish directly the following generalrelation.Lemma 2.31. Remind from (2.11) that we identify y 0 with x and y 1 with yand let 0 j 1 , j 2 , j 3 , k 1 1. Then(2.32)( ) ( )□ k 1− □ k y j 1y j 12y j y 31y j 3y j 2= −1∑k 2 =0C 2B 2□ k 2y j 1y j 2 ·□k 1y j 3y k 2 + 1∑k 2 =0□ k 2y j 1y j 3 ·□k 1y j 2y k 2 .

This lemma is left to the reader; anyway, we shall complete the proof ofa generalization of Lemma 2.31 to the case of m 1 dependent variables(y 1 , . . .,y m ) in Section 2 below (Lemma 3.40).Coming back to the first auxiliary system (2.20), we therefore have obtaineda necessary and sufficient condition for the existence of (X, Y ): thefunctions Π k 1j 1 ,jshould satisfy the following system of first order partial differentialequations, just obtained from (2.30) by replacing the square func-2tions by the Pi functions:(2.33) ⎧( )Π0 − ( )0,0 Π 0 y 0,1 = − x Π1 0,0 · Π0 1,1 + Π1 0,1 · Π0 0,1( ),⎪⎨ Π0 − ( )0,1 Π 0 y 1,1 = − x Π0 0,1 · Π 0 0,1 − Π 1 0,1 · Π 0 1,1 + Π 0 1,1 · Π 0 0,0 + Π 1 1,1 · Π 0 0,1,( )Π1 − ( )0,0 Π 1 y 0,1 = − x Π0 0,0 · Π1 0,1 − Π1 0,0 · Π1 1,1 + Π0 0,1 · Π1 1,1 + Π1 0,1 · Π1 0,1 ,⎪⎩( )Π1 − ( )0,1 Π 1 y 1,1 = − x Π0 0,1 · Π1 0,1 + Π0 1,1 · Π1 0,0 .172.34. Second auxiliary system. It is now time to come back to the functionsG, H, L and M and to get rid of the auxiliary “Pi” functions. Unfortunately,we cannot invert directly the linear system (2.18), hence we must choose twospecific square functions as principal unknowns, and the best, from a combinatorialpoint of view, is to choose □ 0 xx and □ 1 yy. Remind that by (2.20),we have □ 0 xx = Π 0 0,0 and □ 1 yy = Π 1 1,1. For clarity, it will be useful to adoptthe notational equivalences(2.35) Θ 0 ≡ Π 0 0,0 and Θ 1 ≡ Π 1 1,1.We may therefore quasi-inverse the linear system (2.18), obtaining that thefour functions Π 1 0,0 , Π1 0,1 , Π0 0,1 and Π0 1,1 may be expressed in terms of thefunctions G, H, L and M and in terms of the remaining two principal unknowns(2.35), which yields:⎧Π 1 0,0 = □1 xx = −G(2.36)⎪⎨ Π 1 0,1 = □1 xy = −1 2 H + 1 2 Θ0 ,Π 0 0,1⎪⎩= □0 xy = 1 2 L + 1 2 Θ1 ,Π 0 1,1 = □0 yy = M.Replacing now each of these four expressions in the compatibility conditionsof the first auxiliary system (2.33), solving the four equations with respectto Θ 1 y , Θ0 x , Θ1 x and Θ0 y , we get after hygienic simplifications what we shallcall the second auxiliary system, which is a complete system of first order

18partial derivatives in the remaining two principal unknowns Θ 0 and Θ 1 :(2.37) ⎧Θ 1 y = − L y + 2M x + HM − 1 2 L2 + M Θ 0 + 1 ( ) Θ1 2,2⎪⎨ Θ 0 x = − 2 G y + H x + G L − 1 2 H2 − G Θ 1 + 1 2(Θ0 )2 ,Θ 1 x = − 2 3 H y + 1 3 L x + 2 G M − 1 2 H L − 1 2 H Θ1 + 1 2 L Θ0 + 1 2 Θ0 Θ 1 ,⎪⎩ Θ 0 y = − 1 3 H y + 2 3 L x + 2 G M − 1 2 H L − 1 2 H Θ1 + 1 2 L Θ0 + 1 2 Θ0 Θ 1 .We do not comment the intermediate computations, since they offer no newcombinatorial discovery.2.38. Precise lexicographic rules. We group first order derivatives beforezeroth order derivatives; in each group, we respect the lexicographic orderof appearance given by the sequence G, H, L, M, Θ 0 , Θ 1 ; we always put rationalcoefficient of every differential monomial in its left; consequently, weaccept a minus sign just after an equality sign, as for instance in (2.36) 1 andin (2.37) 2 ; for clarity, we prefer to write a complicated differential equationas 0 = Φ, with 0 on the left, instead of Φ = 0, since Φ may incoporate 10,20 and up to 150 monomials, as will happen for instance in the next sectionsbelow.2.39. Compatibility conditions for the second auxiliary system. Clearly,the necessary and sufficient condition for the existence of solutions (Θ 0 , Θ 1 )to the second auxiliary system (2.37) is that the two cross differentiationsvanish:(2.40){ 0 =(Θ0x))− (Θ 0 y y0 = ( )Θ 1 x − ( )Θ 1 y yUsing (2.37), we shall see that we exactly obtain the two second order partialdifferential equations written in Theorem 1.2 (4). For completeness, weshall perform completely the computation of the first compatibility condition(2.40) and leave the second as an (easy) exercise.First of all, inserting (2.37) and using the rule of Leibniz for the differentiationof a product, let us write the crude result, performing neither anyx ,x .

simplification nor any reordering:(2.41) ⎧0 = ( )Θ 0 x − ( )Θ 0 y y x⎪⎨ = − 2 G yy + H xy + G y L + G L y − H H y − G y Θ 1 − G Θ 1 y + Θ 0 Θ 0 y++ 1 3 H xy − 2 3 L xx − 2 G x M − 2 G M x + 1 2 H x L + 1 2 H L x+19⎪⎩+ 1 2 H x Θ 1 + 1 2 H Θ1 x − 1 2 L x Θ 0 − 1 2 L Θ0 x − 1 2 Θ0 x Θ1 − 1 2 Θ0 Θ 1 x .Next, replacing each first order derivative Θ 0 x, Θ 0 y, Θ 1 x and Θ 1 y occuringin (2.41) by its expression given in (2.37), we obtain (suffering a little) as abrute result, before any simplification (except that we put all second orderderivatives in the beginning):(2.42) ⎧0 = − 2 G yy + 4 3 H xy − 2 3 L xx++ G y L + G L y − H H y − G y Θ 1 + G L y − 2 G M x − G H M++ 1 2 G (L)2 − G M Θ 0 − 1 2 G (Θ1 ) 2 − 1 3 H y Θ 0 + 2 3 L x Θ 0 ++ 2 G M Θ 0 − 1 2 H L Θ0 − 1 2 H Θ0 Θ 1 + 1 2 L (Θ0 ) 2 + 1 2 (Θ0 ) 2 Θ 1 −⎪⎨− 2 G x M − 2 G M x + 1 2 H x L + 1 2 H L x + 1 2 H x Θ 1 ++ 1 6 H L x − 1 3 H H y + G H M − 1 4 (H)2 L − 1 4 (H)2 Θ 1 ++ 1 4 H L Θ0 + 1 4 H Θ0 Θ 1 − 1 2 L x Θ 0 + G y L − 1 2 H x L − 1 2 G (L)2 ++ 1 4 H2 L + 1 2 G L Θ1 − 1 4 L ( Θ 0)2 + G y Θ 1 − 1 2 H x Θ 1 −− 1 2 G L Θ1 + 1 4 H2 Θ 1 + 1 2 G (Θ1 ) 2 − 1 4(Θ0 )2 Θ 1 − 1 6 L x Θ 0 ++ 1 3 H y Θ 0 − G M Θ 0 + 1 4 H L Θ0 + 1 4 H Θ0 Θ 1 − 1 4 L (Θ0 ) 2 −⎪⎩− 1 4 (Θ0 ) 2 Θ 1 .

20Now, we can simplify this brute expression by chasing every couple (ortriple, or quadruple) of terms killing each other. After (patient) simplificationand lexicographic ordering, we obtain the equation(2.43) ⎧⎪⎨ 0 = −2 G yy + 4 3 H xy − 2 3 L xx+⎪⎩+ 2(G L) y − 2 G x M − 4 G M x + 2 3 H L x − 4 3 H H y,which is exactly the first equation of (4) of Theorem 1.2. The treatment ofthe second one is totally similar. This completes the proof of the equivalencebetween (1) and (4) in Theorem 1.2.2.44. Interlude: about hand-computed formulas. In Section 4 below,when dealing with several dependent variables y 1 , . . .,y m , many simplificationsof identities which are much more massive than (2.42) will occurseveral times. It is therefore welcome to explain how we manage to achievesuch computations, without mistakes at the end and strictly by hand. One ofthe trick is to use colors, which, unfortunately, cannot be restituted in thisprinted document. Another trick is to underline and to number the termswhich disappear together, by pair, by triple, by quadruple, etc. This trickis illustrated in the detailed identity (2.45) below, extracted from our manuscript,which is a copy of (2.42) together with the designation of all theterms which vanish together. Hence, we keep a written track of each intermediatestep of every computation and of every simplification. Checking thecorrectness of a computation simply by reading is then the easiest way, bothfor the writer and for the reader, although of course, it takes time, anyway.On the contrary, when relying upon a digital computer, most intermediatesteps are invisible; the chase of mistakes is by reading the program andby testing it on several instances. Alas, all the finest intuitions which mayawake in the extreme inside of a long computation are essentially absent, themind believing that the machine is stronger for such tasks. This last belief isin part true, in the case where straightforward known computations are concerned,and in part untrue, in the case where some new hidden mathematicalreality is concerned.For us, the challenge is to control everything in a sea of signs. Computationsare to be organized like a living giant coral tree, all part of whichshould be clearly visible in a transparent fluid of thought, and permanentlysubject to corrections.Indeed, it often happens that going through a problem involving massiveformal computations, some disharmony or some incoherency is discovered.Then one has to inspect every living atom in the preceding branches of the

aleza, y por otra, las organizaciones necesarias para fijar las relacionesentre ellos. Cultura es todo aquello que incluye el conocimiento, lascreencias, el arte, la moral, el derecho, las costumbres y cualquier otrohábito y capacidades adquiridos por el hombre. Es un sistema de significadoe información compartido por un grupo y transmitido a travésde generaciones, que permite proveer de sus necesidades básicas desupervivencia coordinada a la conducta social para lograr una existenciaviable. Es algo creado por el hombre frente a la naturaleza. La culturaes la herencia social, es la manera en que los seres humanos solucionanproblemas y a su vez un conjunto de actitudes, creencias, valores,hábitos, tradiciones adquiridos por los miembros de la sociedad.Choque cultural• El choque cultural es un término que fue introducido por primeravez en 1958, describe la ansiedad producida por la pérdida delsentido de qué hacer, cuándo hacer, o cómo hacer las cosas enun nuevo ambiente. Este término expresa un sentimiento de faltade dirección porque el individuo no conoce qué es apropiado oinapropiado en el nuevo lugar. Se presenta generalmente despuésde las primeras semanas de haber llegado a un nuevo sitio.• Podemos describir el Choque cultural como la incomodidad físicay emocional que se sufre cuando se llega a vivir a un lugar diferenteal de origen. Las conductas aprendidas no son aceptadas o[ 21 ]

22summed together. However, whereas we use the red pencil to underline thevanishing terms, we use the green pencil to underline the remaining terms.This small trick is to avoid as much as possible to copy several times somelong formal expressions. Finally, we reorder everything lexicographically,so as to get the conclusion (2.43). In order to obtain the final equation (2.43)as efficiently as possible, we read the remaining terms, picking them directlyin lexicographic order. If, by lack of luck, one or two terms are forgotten bythe eyes and not written in the right place, we copy once more the very finalresult in the right order, or we use the blank corrector.Of course, such a refined methodology could seem to be essentially superfluousfor such relatively accessible computations. However, when passingto several dependent variables, the current expressions will be approximativelyfive times more massive. We may really ascertain that a clevermethodology of hand computations is helpful in this category.§3. SYSTEMS OF SECOND ORDER ORDINARY DIFFERENTIALEQUATIONS EQUIVALENT TO FREE PARTICLES3.1. Combinatorics of the second order prolongation of a point transformation.In this section, we endeavour to explain how Lie’s theoremand proof may be generalized to the case of several dependent variables.As in the statement of Theorem 1.7 (1), let us assume that the systemyxx j = F j (x, y, y x ), j = 1, . . .,m, is equivalent under an invertible pointtransformation (x, y) ↦→ (X(x, y), Y (x, y)) to the free particle systemY j XX= 0, j = 1, . . .,m. By assumption, the Jacobian determinant∣ X x X y 1 . . . X y m ∣∣∣∣∣∣∣(3.2) ∆(x|y 1 | . . . |y m ) :=Yx 1 Y 1y. . . Y 1∣1 y m. . . . . . . . . . . .∣Y mxY my 1 . . . Y my mdoes not vanish at the origin. As in the case m = 1, since the flat systemY j XX= 0 is left unchanged by any affine transformation, we can (and weshall) assume that the transformation is tangent to the identity at the origin,so that the above Jacobian matrix equals the identity matrix at (x, y) =(0, 0), whence in a neighborhood of the origin it is close to the identitymatrix, namely(3.3) X x∼ = 1, Xy j ∼ = 0, Y jx ∼ = 0, Y j 2y j 1∼ = δj 2j 1.Inductive formulas for the computation how the differential equation in the(X, Y ) coordinates is related to the differential equation in the (x, y) coordinatesmay be found in [BK1989], [Ol1995]; the explicit formulas are notachieved in these references. Let us recall the inductive formulas, just on the

computational level (differential-geometric conceptional background aboutgraph transformations may be found in [Ol1986], Ch. 2).First of all, we seek how the Y j dY jX:= are explicitely related to the dX yl x .It suffices to replace, in the identity(3.4) ()m∑mY j X · X x dx + X y l dy l = Y j X dX = dY j = Yx j dx + ∑Y j dy ly ll=1the differentials dy l by y l x dx and then to identify the coefficient of dx onboth sides, which rapidly yields the formulas(3.5) Y j X = Y x j + ∑ ml=1 yl x Y jy lX x + ∑ ml=1 yl x X ,y lfor j = 1, . . .,m.Next, we seek how the Y j XX := d2 Y jdX 2l=123= dY j XdX are related to the yl 1 x , y l 2 xx .It suffices to again replace each dy l by y l x dx and each dy l x by y l xx dx in theidentity(3.6) ⎧⎪⎨⎪⎩Y j XX ·(X x dx +)m∑X y l dy l = Y j XX · dX = dY j Xl=1= ∂Y j mX∂x dx + ∑l=1(∂Y j mX=∂x + ∑l=1∂Y j X∂y l dy l +∂Y j X∂y l y l x +m∑l=1m∑l=1∂Y j X∂y l x∂Y j X∂y l xBefore entering the precise combinatorics of the explicit expression of Y j XX ,let us observe that the last term of (3.6) simply writes D(Y j X) dx, where Ddenotes the total differentiation operator (of order two) defined by(3.7) D := ∂∂x + m∑l=1y l x∂∂y + ∑ m ll=1∂yxxl ∂yxlSince dX ≡ DX after replacing each dy l by yx l dx, it follows that we maycompactly rewrite (3.6) as(3.8) Y j XX DX · dx = D ( Y j X)· dxConsequently, the expressions of Y j X (obtained in (3.5)) and of Y j XX are(3.9)Y j X = DY jand Y j XXDX= D ( Y j )XDX= DDY j · DX − DDX · DY j.[DX] 3.y l xxdy l x)· dx.

24As, by assumption, the system yxx j = F j (x, y, y x ) transforms to the flatsystem Y j XX= 0, after erasing the denominator of (3.7), we come to theequations(3.10) 0 = DDY j · DX − DDX · DY j ,for j = 1, . . ., m. However, this too simple and too compact expression ofthe system y j xx = F j (x, y, y x ) is of no use and we must develope (patiently!)the explicit expressions of DDY j , of DX, of DDX and of DY j , using thecomplete expression of D defined in (3.6).At this point, we would like to stress that it constitutes already a nontrivialcomputational and combinatorial task to obtain a complete explicit formulafor the system y j xx = F j (x, y, y x ) hidden in the compact form (3.10), whichwould be the generalization of the nice formula (2.9) involving modificationsof the Jacobian determinant. For general m 2, the complete proofsare postponed to Section 5 below.Since it would be intuitively unsatisfactory to provide directly the finalsimplified expression of the development of (3.10) in the general casem 2, let us firstly describe step by step how one may guess what is thegeneralization of (2.9).

For instance, in the case m = 2, by a direct and relatively short computationwhich consists in developing plainly (3.10), we obtain for j = 1, 2:(3.11)⎧0 = − X xx Yx j + Y xx j X x+[]+ yx 1 · −X xx Y jy+ Y j1 xx X y 1 − 2 X xy 1 Y x j + 2 Y jxyX 1 x +[+ yx 2 · −X xx Y jy+ Y j2 xx X y 2 − 2 X xy 2 Y x j + 2Y jxyX 2 x]+[]+ yx 1 yx 1 · −2 X xy 1 Y jy+ 2 Y j1 xyX 1 y 1 − X y 1 y 1 Y x j + Y jy 1 yX 1 x +[+ yx 1 y2 x · −2 X xy 1 Y jy+ 2 Y j2 xyX 1 y 2 − 2 X xy 2 Y jy+ 2 Y j1 xyX 2 y 1−⎪⎨⎪⎩]−2 X y 1 y 2 Y x j + 2 Y jy 1 yX 2 x +[]+ yx 2 yx 2 · −2 X xy 2 Y jy+ 2 Y j2 xyX 2 y 2 − X y 2 y 2 Y x j + Y jy 2 yX 2 x +[]+ yx 1 y1 x y1 x · −X y 1 y 1 Y jy+ Y j1 y 1 yX 1 y 1 +[]+ yx 1 y1 x y2 x · −X y 1 y 1 Y jy+ Y j2 y 1 yX 1 y 2 − 2 X y 1 y 2 Y jy+ 2 Y j1 y 1 yX 2 y 1 +[]+ yx 1 yx 2 yx 2 · −X y 2 y 2 Y jy+ Y j1 y 2 yX 2 y 1 − 2 X y 1 y 2 Y jy+ 2 Y j2 y 1 yX 2 y 2 +[]+ yx 2 yx 2 yx 2 · −X y 2 y 2 Y jy+ Y j2 y 2 yX 2 y 2 +[{}]+ yxx 1 · −X y 1 Yx j + Y jyX 1 x + yx 2 · −X y 1 Y jy+ Y j2 yX 1 y 2 ++ y 2 xx ·[−X y 2 Y jx + Y jy 2 X x + y 1 x ·{−X y 2 Y jy 1 + Y jy 2 X y 1}].Unfortunately, the above two equations are not solved with respect to yxx1and to yxx 2 . Consequently, if we abbreviate them as a linear system of theform{0 = A 1 + yxx 1 · B1 1 + y2 xx · B1 2 ,(3.12)0 = A 2 + y 1 xx · B2 1 + y2 xx · B2 2 ,we have to solve for yxx 1 and for yxx 2 by means of the classical rule of Cramer.Here, it is rather quick to check manually that the determinant of this systemhas the following nice expression:⎧⎪⎨(3.13)∣ B1 1 B21 B1 2 B22 ∣ = ∆(x|y1 |y 2 ) · {Xx + yx 1 X y 1 + y2 x X }y 2⎪⎩= ∆(x|y 1 |y 2 ) · DX.However, the complete solving for yxx 1 and for y2 xx requires some more time.After a direct and rather long hand computation (or alternately, using Maple25

26or Mathematica) one obtains formulas involving hidden 3 × 3 determinants,which have to be guessed by the intuition; the first equation that we obtain,namely for yxx 1 is as follows:(3.14)∣ ∣ X x X y 1 X y 2 ∣∣∣∣∣ X x X xx X y 2 ∣∣∣∣∣0 = yxx 1 ·Yx 1 Y 1yY 1∣1 y +2YYx 2 Y 2yY 2x 1 Yxx 1 Y 1y +∣21 yY 22 x Yxx 2 Y 2y⎧ ∣ ∣ 2 ∣⎫ ⎨ ∣∣∣∣∣ X x X xy 1 X y 2 ∣∣∣∣∣ + yx 1 · ⎩ 2 Y x 1 Y X xx X y 1 X y 2 ∣∣∣∣∣ ⎬ 1xyY 1 1 y −2Y 1Yx 2 Y 2xyY 2xx Y 1yY 1∣1 y 21 yY 22 xx Y 2yY 2 ⎭ +1 y⎧ ∣ 2⎨ ∣∣∣∣∣ X x X xy 2 X y 2+ yx 2 · ⎩ 2 Y x 1 Y 1xyY 1 2 y 2Yx 2 Yxy 2 Y 2 2y 2 ∣ ∣∣∣∣∣⎫⎬⎭ +⎧⎨+ yx 1 y1 x ·X x X y 1 y 1 X ∣ ∣⎫ y ∣∣∣∣∣ 2X xy 1 X y 1 X y 2 ∣∣∣∣∣ ⎬ ⎩Yx 1 Y 1y∣1 yY 1 1 y − 22Y 1Yx 2 Y 2y 1 yY 2xyY 1∣1 yY 1 1 y 21 yY 22 xyY 2 1 yY 2 ⎭ +1 y⎧ ∣ 2⎨ ∣∣∣∣∣ X x X y 1 y 2 X ∣ ∣⎫ y ∣∣∣∣∣ 2+ yx 1 yx2 ⎩ 2 Y x 1 Y X xy 2 X y 1 X y 2 ∣∣∣∣∣ ⎬ 1y 1 yY 1 2 y − 22Y 1Yx 2 Y 2y 1 yY 2xyY 1∣2 yY 1 1 y 22 yY 22 xyY 2 2 yY 2 ⎭ +1 y⎧2⎨+ yx 2 y2 x ·X x X y 2 y 2 X ∣⎫ y ∣∣∣∣∣ 2 ⎬ ⎩Yx 1 Y 1y∣2 yY 1 2 y 2Yx 2 Y 2y 2 yY 2 ⎭ +2 y 2⎧ ∣⎨ ∣∣∣∣∣ X y 1 y 1 X y 1 X ∣⎫ ⎧ ∣y ∣∣∣∣∣ 2 ⎬ ⎨ ∣∣∣∣∣ X y 1+ yx 1 yx 1 yx 1 ·⎩ − Y 1y1y Y 1 1 yY 1y 2 X y 1 X ∣⎫ y ∣∣∣∣∣ 2 ⎬ 1 y 2Y 2y 1 yY 2 1 yY 2 ⎭ + y1 x yx 1 yx 2 ·⎩ −2 Y 1y1y Y 1 2 yY 1 1 y 21 yY 22 y 1 yY 2 2 yY 2 ⎭ +1 y⎧ ∣ 2⎨ ∣∣∣∣∣ X y 2 y 2 X y 1 X y 2+ yx 1 yx 2 yx 2 ·⎩ − Y 1y2y Y 1 2 yY 1 1 y 2Y 2y 2 yY 2 2 yY 2 1y 2 ∣ ∣∣∣∣∣⎫⎬⎭ .This formula and the next have been checked by Sylvain Neut and MichelPetitot with the help of Maple. We notice that the size is not negligible,but fortunately, there appears some combinatorics, much more visible than

in (3.11). The second equation that we obtain, namely for y 2 xx, is as follows:27(3.15)∣ 0 = yxx 2 ·X x X y 1 X y 2 ∣∣∣∣∣ X x X y 1 X xxYx 1 Y 1yY 1∣1 y +2YYx 2 Y 2yY 2x 1 Y 1yY 1∣1 xx1 yY 22 x Y 2yY 2 ∣ +2 xx⎧ ∣ ∣⎫ ⎨ ∣∣∣∣∣ X x X y 1 X xy 1 ∣∣∣∣∣ ⎬+ yx 1 ·⎩ 2 Y x 1 Y 1yY 1 1 xy 1Yx 2 Y 2yY 2 ⎭ +1 xy⎧ ∣ 1 ∣ ∣⎫ ⎨ ∣∣∣∣∣ X x X y 1 X xy 2 ∣∣∣∣∣ + yx 2 ·⎩ 2 Y x 1 Y X xx X y 1 X y 2 ∣∣∣∣∣ ⎬ 1yY 1 1 xy −2Y 1Yx 2 Y 2yY 2xx Y 1yY 1∣1 y 21 xyY 22 xx Y 2yY 2 ⎭ +1 y 2⎧∣⎫ ⎨+ yx 1 y1 x ·X x X y 1 X y 1 y ∣∣∣∣∣ 1 ⎬ ⎩Yx 1 Y 1yY 1∣1 y 1 y 1Yx 2 Yy 2 Y 2 ⎭ +1 y 1 y⎧ ∣ 1 ∣ ∣⎫ ⎨ ∣∣∣∣∣ X x X y 1 X y 1 y ∣∣∣∣∣ 2+ yx 1 yx2 ⎩ 2 Y x 1 Y X xy 1 X y 1 X y 2 ∣∣∣∣∣ ⎬ 1yY 1 1 y 1 y − 22Y 1Yx 2 Y 2yY 2xyY 1∣1 yY 1 1 y 21 y 1 yY 22 xyY 2 1 yY 2 ⎭ +1 y⎧∣ 2 ∣⎫ ⎨X x X y 1 X y 2 y ∣∣∣∣∣ 2X xy 2 X y 1 X y 2 ∣∣∣∣∣ ⎬+ yx 2 yx 2 ·⎩Yx 1 Y 1yY 1∣1 y 2 y − 22Y 1Yx 2 Y 2yY 2xyY 1∣2 yY 1 1 y 21 y 2 yY 22 xyY 2 2 yY 2 ⎭ +1 y 2⎧ ∣⎨ ∣∣∣∣∣ X y 1 y 1 X y 1 X ∣⎫ ⎧ ∣y ∣∣∣∣∣ 2 ⎬ ⎨ ∣∣∣∣∣ X y 1+ yx 1 yx 1 yx 2 ·⎩ − Y 1y1y Y 1 1 yY 1y 2 X y 1 X ∣⎫ y ∣∣∣∣∣ 2 ⎬ 1 y 2Y 2y 1 yY 2 1 yY 2 ⎭ + y1 x yx 2 yx 2 ·⎩ −2 Y 1y1y Y 1 2 yY 1 1 y 21 yY 22 y 1 yY 2 2 yY 2 ⎭ +1 y⎧ ∣ 2⎨ ∣∣∣∣∣ X y 2 y 2 X y 1 X y 2+ yx 2 yx 2 yx 2 ·⎩ − Y 1y2y Y 1 2 yY 1 1 y 2Y 2y 2 yY 2 2 yY 2 1y 2 ∣ ∣∣∣∣∣⎫⎬⎭ .Importantly, the obtained formulas seem to be analogous to the formula(2.9), since we observe that the coefficients of the degree three polynomialin the y l x are modifications of the Jacobian determinant ∆(x|y1 |y 2 ).To describe the underlying combinatorics, let us observe that there existexactly six possible distinct second order derivatives: xx, xy 1 , xy 2 , y 1 y 1 ,y 1 y 2 and y 2 y 2 . There are also exactly three columns in the Jacobian determinant(3.2). By replacing each of the three columns of first order derivativesby a column of second order detivatives (leaving X, Y 1 and Y 2 unchanged),

28we may build exactly eighteen new determinants(3.16)⎧⎪⎨⎪⎩∆(xx|y 1 |y 2 ) ∆(x|xx|y 2 ) ∆(x|y 1 |xx)∆(xy 1 |y 1 |y 2 ) ∆(x|xy 1 |y 2 ) ∆(x|y 1 |xy 1 )∆(xy 2 |y 1 |y 2 ) ∆(x|xy 2 |y 2 ) ∆(x|y 1 |xy 2 )∆(y 1 y 1 |y 1 |y 2 ) ∆(x|y 1 y 1 |y 2 ) ∆(x|y 1 |y 1 y 1 )∆(y 1 y 2 |y 1 |y 2 ) ∆(x|y 1 y 2 |y 2 ) ∆(x|y 1 |y 1 y 2 )∆(y 2 y 2 |y 1 |y 2 ) ∆(x|y 2 y 2 |y 2 ) ∆(x|y 1 |y 2 y 2 ),where for instance(3.17)⎧∆(y 1 y 2 |y 1 |y 2 ) :=⎪⎨∆(x|y⎪⎩1 |xy 2 ) :=∣∣X y 1 y 2 X y 1 X ∣y ∣∣∣∣∣∣ 2Y 1y 1 yY 1 2 yY 1 1 y 2 andY 2y 1 yY 2 2 yY 2 1 y 2∣X x X y 1 X xy 2 ∣∣∣∣∣∣Yx 1 Y 1yY 1 1 xy 2 .Yx 2 Y 2yY 2 1 xy 2Hence, using the ∆-notation, we may rewrite the two equation (3.14)and (3.15) under a more compact form; after division by the Jacobian determinant∆(x|y 1 |y 2 ), the first equation becomes:(3.18)⎧{}0 = yxx 1 + ∆(x|xx|y2 )∆(x|y 1 |y 2 ) + y1 x · 2 ∆(x|xy1 |y 2 )∆(x|y 1 |y 2 ) − ∆(xx|y1 |y 2 )+∆(x|y 1 |y 2 ){ } { }+ yx 2 · 2 ∆(x|xy2 |y 2 ) ∆(x|y+ y 1∆(x|y⎪⎨1 |y 2 x yx 1 1 y 1 |y 2 )·− 2 ∆(xy1 |y 1 |y 2 )+) ∆(x|y 1 |y 2 ) ∆(x|y 1 |y 2 ){} { }+ yx 1 y2 x · 2 ∆(x|y1 y 2 |y 2 )− 2 ∆(xy2 |y 1 |y 2 ) ∆(x|y+ y 2 2∆(x|y 1 |y 2 ) ∆(x|y 1 |y 2 x)y2 x · y 2 |y 2 )+∆(x|y 1 |y 2 ){} {}+ yx 1 y1 x y1 x · − ∆(y1 y 1 |y 1 |y 2 )+ y 1∆(x|y 1 |y 2 x)y1 x y2 x · −2 ∆(y1 y 2 |y 1 |y 2 )+∆(x|y 1 |y 2 ){}⎪⎩ + yx 1 yx 2 yx 2 · − ∆(y2 y 2 |y 1 |y 2 ).∆(x|y 1 |y 2 )

Similarly, the second equation takes the form:(3.19) ⎧{ }0 = yxx 2 + ∆(x|y1 |xx)∆(x|y 1 |y 2 ) + y1 x · 2 ∆(x|y1 |xy 1 )+∆(x|y 1 |y 2 ){}+ yx 2 · 2 ∆(x|y1 |xy 2 )∆(x|y 1 |y 2 ) − ∆(xx|y1 |y 2 )+ y∆(x|y 1 |y 2 x 1 yx 1 ·){}⎪⎨ + yx 1 y2 x · 2 ∆(x|y1 |y 1 y 2 )− 2 ∆(xy1 |y 1 |y 2 )+∆(x|y 1 |y 2 ) ∆(x|y 1 |y 2 ){ }∆(x|y+ yx 2 yx 2 2 |y 2 y 2 )·− 2 ∆(xy2 |y 1 |y 2 )+∆(x|y 1 |y 2 ) ∆(x|y 1 |y 2 ){}+ yx 1 yx 1 yx 2 · − ∆(y1 y 1 |y 1 |y 2 )+ y∆(x|y 1 |y 2 x 1 yx 2 yx 2 ·){}⎪⎩ + yx 2 y2 x y2 x · − ∆(y2 y 2 |y 1 |y 2 ).∆(x|y 1 |y 2 )29{ } ∆(x|y 1 |y 1 y 1 )+∆(x|y 1 |y 2 ){−2 ∆(y1 y 2 |y 1 |y 2 )∆(x|y 1 |y 2 )Since the formulas are still of a consequent size, analogously to what wasachieved in Section 2, we shall introduce a new family of square functions asfollows. We first index the coordinates (x, y 1 , . . .,y m ) as (y 0 , y 1 , . . ., y m ),namely we introduce the notational equivalence(3.20) y 0 ≡ x ,which will be very convenient in the sequel, especially in order to write generalformulas. With this convention at hand, our eighteen square functions□ k 1y l 1y , defined for 0 j l 2 1, j 2 , k 1 2 are defined by(3.21)⎧⎪⎨⎪⎩□ 0 xx := ∆(xx|y1 |y 2 )∆(x|y 1 |y 2 )□ 0 y 1 y 1 := ∆(y1 y 1 |y 1 |y 2 )∆(x|y 1 |y 2 )□ 1 xx := ∆(x|xx|y2 )∆(x|y 1 |y 2 )□ 1 y 1 y 1 := ∆(x|y1 y 1 |y 2 )∆(x|y 1 |y 2 )□ 2 xx := ∆(x|y1 |xx)∆(x|y 1 |y 2 )□ 2 y 1 y 1 := ∆(x|y1 |y 1 y 1 )∆(x|y 1 |y 2 )□ 0 xy 1 := ∆(xy1 |y 1 |y 2 )∆(x|y 1 |y 2 )□ 0 y 1 y 2 := ∆(y1 y 2 |y 1 |y 2 )∆(x|y 1 |y 2 )□ 1 xy 1 := ∆(x|xy1 |y 2 )∆(x|y 1 |y 2 )□ 1 y 1 y 2 := ∆(x|y1 y 2 |y 2 )∆(x|y 1 |y 2 )□ 2 xy 1 := ∆(x|y1 |xy 1 )∆(x|y 1 |y 2 )□ 2 y 1 y 2 := ∆(x|y1 |y 1 y 2 )∆(x|y 1 |y 2 )}+□ 0 xy 2 := ∆(xy2 |y 1 |y 2 )∆(x|y 1 |y 2 )□ 0 y 2 y 2 := ∆(y2 y 2 |y 1 |y 2 )∆(x|y 1 |y 2 )□ 1 xy 2 := ∆(x|xy2 |y 2 )∆(x|y 1 |y 2 )□ 1 y 2 y 2 := ∆(x|y2 y 2 |y 2 )∆(x|y 1 |y 2 )□ 2 xy 2 := ∆(x|y1 |xy 2 )∆(x|y 1 |y 2 )□ 2 y 2 y 2 := ∆(x|y1 |y 2 y 2 )∆(x|y 1 |y 2 )Obviously, the square functions are symmetric with respect to the lowerindices: □ k 1y l 1y = l 2 □k 1. Here, the upper index designates the columny l 2y l 1

30upon which the second order derivative appears, itself being encoded by thetwo lower indices. Even if this is hidden in the notation, we shall rememberthat the square functions are explicit rational expressions in terms of thesecond order jet of the transformation (x, y) ↦→ (X(x, y), Y (x, y)). At thispoint, we may summarize what we have established so far.Lemma 3.22. The system of two second order ordinary differential equationsyxx 1 = F 1 (x, y, y x ) and yxx 2 = F 2 (x, y, y x ) is equivalent, under a pointtransformation, to the flat system YXX 1 = 0 and Y XX 2 = 0 if and only if thereexist three local K-analytic functions X(x, y), Y 1 (x, y) and Y 2 (x, y) suchthat it may be written under the form(3.23)⎧0 = yxx 1 + □ 1 xx + yx 1 · (2□ 1 xy − xx) 1 □0 + y2x · (2 )□ 1 xy + 2⎪⎨⎪⎩+ y 1 x y 1 x · (□ 1 y 1 y 1 − 2 □0 xy 1 )+ y1x y 2 x · (2□ 1 y 1 y 2 − 2 □0 xy 2 )+ y2x y 2 x · (□ 1 y 2 y 2 )++ y 1 x y 1 x y 1 x · (−□ 0 y 1 y 1 )+ y1x y 1 x y 2 x · (−2□ 0 y 1 y 2 )+ y1x y 2 x y 2 x · (−□ 0 y 2 y 2 ),0 = y 2 xx + □ 2 xx + y 1 x · (2□ 2 xy 1 )+ y2x · (2□ 2 xy 2 − □0 xx)++ y 1 x y 1 x · (□ 2 y 1 y 1 )+ y1x y 2 x · (2□ 2 y 1 y 2 − 2 □0 xy 1 )+ y2x y 2 x · (□ 2 y 2 y 2 − 2 □0 xy 2 )++ y 1 x y 1 x y 2 x · (−□ 0 y 1 y 1 )+ y1x y 2 x y 2 x · (−2□ 0 y 1 y 2 )+ y2x y 2 x y 2 x · (−□ 0 y 2 y 2 ).3.24. Second Lie prolongation of a vector field. At this point, instead ofproceeding further with the case m = 2, it is now time to pass to the generalcase m 2. First of all, we would like to remind from [GM2003] thecomplete explicit expression of the point prolongation to the second orderjet space of a general vector field of the form L = X ∂ + ∑ m∂x j=1 Y j ∂ : it∂y jis a vector field of the form(3.25) L (2) = X ∂∂x + m∑j=1Y j∂∂y + ∑ m jj=1∂R j 1∂yxj+m∑j=1∂R j 2∂yxxj,where the coefficients R j 1 and R j 2 are polynomials in the jet space variableshaving as coefficients certain specific linear combinations of first and second

order derivatives of X and of the Y j :(3.26) ⎧⎪⎨⎪⎩R j 1 = Y jx + m∑l 1 =1R j 2 = Y jxx + m∑++++m∑l 1 =1m∑l 1 =1 l 2 =1m∑m∑l 1 =1 l 2 =1 l 3 =1m∑l 1 =1m∑y l 1xx ·m∑l 1 =1 l 2 =1[ ]y l 1x · Y j − y l 1 δj l 1X x +m∑[]y l 1x · 2 Y j − xy l 1 δj l 1X xx +m∑l 1 =1 l 2 =1y l 1x y l 2x · [−δ j l 1X y l 2],[]y l 1x y l 2x · Y jy l 1y − l 2 δj l 1X xy l 2 − δ j l 2X xy l 1 +m∑y l 1x y l 2x y l 3x · [−δ j l 1X y l 2y l 3]+[Y jy l 1 − 2 δj l 1X x]+y l 1x y l 2xx · [−δ j l 1X y l 2 − 2 δ j l 2X y l 1].However, since the notations in [GM2003] are different and since the generalcase of n 1 independent variables and m 1 dependent variables isconsidered there, it is certainly easier to reconstiture formulas (3.26) directlyby means of the inductive formulas described in [Ol1986], [BK1989]).Analogously to the observation made in Section 2, we guess that thereexists a formal correspondence between the terms of R j 2 not involving yl xxand the explicit form of the equation yxx j = F j (x, y, y x ) equivalent to Y j XX =0. In the case m = 2, we claim that this formal correspondence also holdstrue. Indeed, it suffices to write formula (3.26) for R j 2 modulo the yxx, l whichyields two expressions in total analogy with the two explicit polynomialsappearing in the right-hand side of (3.23):(3.27)⎧⎪⎨⎪⎩R 1 2 (mod y l xx) ≡ Y 1xx + y 1 x · {2Y 1xy 1 − X xx31}+ y2x · {2Y 1xy 2 }+ y1x y 1 x · {Y 1y 1 y 1 − 2 X xy 1 }++ yx 1 y2 x · {2Y 1y 1 y − 2 X } 2 xy 2 + y2x yx 2 · {Y } 1y 2 y + 2+ yx 1 y1 x y1 x · {−X y 1 y 1} + y1 x y1 x y2 x · {−2 X y 1 y 2} + y1 x y2 x y2 x · {−X y 2 y 2} ,R 2 2 (mod y l xx) ≡ Y 2xx + y 1 x · {2Y 2xy 1 }+ y2x · {2Y 2xy 2 − X xx}+ y1x yx 1 · {Y } 2y 1 y + 1+ y 1 x y2 x · {2Y 2y 1 y 2 − 2 X xy 1 }+ y2x y 2 x · {Y 2y 2 y 2 − 2 X xy 2 }++ y 1 x y1 x y2 x · {−X y 1 y 1} + y1 x y2 x y2 x · {−2 X y 1 y 2} + y2 x y2 x y2 x · {−X y 2 y 2} ,Except for inductive inspiration (see the formulation of Lemma 3.32 below),this observation will not be used further. At this stage, it helps at least

32to maintain a strong intuitive control of the correctness of the underlyingcombinatorics.3.28. System equivalent to the flat system. By induction, we thereforeguess that the analogy holds for general m 2, namely we guess the followingcombinatorics, which requires some preliminaries.As in the beginning of §3.1, let x ∈ K, let y = (y 1 , . . .,y m ) ∈ K m , let(x, y) ↦→ (X(x, y), Y (x, y)) be a local K-analytic transformation defined ina neighborhood of the origin in K m+1 and assume that the system y j xx =F j (x, y, y x ), j = 1, . . .,m, is equivalent to the flat system Y j XX = 0, j =1, . . ., m. By assumption, the Jacobian matrix of the equivalence equals theidentity matrix at the origin. Remind that we identify x with y 0 . For allk 1 , l 1 , l 2 = 0, . . ., m, we define a modification(3.29) ∆(x| . . . | k 1y l 1y l 2| . . . |y m )of the Jacobian determinant as follows. We replace the k 1 -th column of thedeterminant (3.2), which consists of first order derivatives ·yk 1 , by a columnwhich consists of second order derivatives ·yl 1y l 2 . In (3.29), the notation | k 1designates the k 1 -th column, the first one being labelled by k 1 = 0 and thelast one by k 1 = m. With this notation at hand, we may define the squarefunctions(3.30) □ k 1y l 1y l 2 := ∆(x| . . . |k 1y l 1y l 2| . . . |y m )∆(x| . . . | k 1 yk 1| . . .ym ) ,which are rational expressions in the second order jet of the transformation(x, y) ↦→ (X(x, y), Y (x, y)). As before, the denominator is the Jacobiandeterminant of the change of coordinates.Since, according to (3.26), the expression of R j 2 (mod yl xx ) is(3.31) ⎧⎪⎨⎪⎩R j 2 (mod y l xx) = Y jxx +++m∑l 1 =1m∑l 1 =1 l 2 =1m∑[]y l 1x · 2 Y j − xy l 1 δj l 1X xx +m∑m∑l 1 =1 l 2 =1 l 3 =1[]y l 1x y l 2x · Y jy l 1y − l 2 δj l 1X xy l 2 − δ j l 2X xy l 1 +m∑y l 1x y l 2x y l 3x · [−δ j l 1X y l 2y l 3],and since, in the cases m = 1 and m = 2, we have already observed stronganalogies between (3.31) and the complete explicit expression of the systemyxx j = F j (x, y, y x ) equivalent to the flat system Y j XX= 0, we guess that thefollowing lemma is formally true.

Lemma 3.32. The system yxx j = F j (x, y, y x ), j = 1, . . .,m, is equivalentto the flat system Y j XX= 0, j = 1, . . .,m, if and only if there exist localK-analytic functions X(x, y) and Y j (x, y), j = 1, . . .,m, such that it maybe written under the specific form(3.33) ⎧⎪⎨⎪⎩0 = y j xx + □j xx + m∑+l 1 =1m∑l 1 =1 l 2 =1+ y j x[]y l 1x · 2 □ j − xy l 1 δj l 1□ 0 xx +m∑m∑l 1 =1 l 2 =1[]y l 1x y l 2x · □ j y l 1y − l 2 δj l 1□ 0 xy − l 2 δj l 2□ 0 xy+ l 1m∑[ ]y l 1x y l 2x · −□ 0 y l 1y. l 2The complete proof of this lemma involves only linear algebra considerations,although with rather massive terms. This makes it rather lengthy.Consequently, we postpone it to the final Section 5 below.3.34. First auxiliary system. Clearly, if we set⎧G j := −□ j xx,⎪⎨ H j l 1:= −2 □ j(3.35)+ xy l 1 δj l 1□ 0 xx ,L j l 1 ,l 2:= −□ j y⎪⎩l 1y + l 2 δj l 1□ 0 xy l 2+ δ j l 2□ 0 xy 1, lM l1 ,l 2:= □ 0 y l 1y , l 2we immediately see that the first condition of Theorem 1.7 holds true. Moreover,we claim that there are m+1 more square functions than functions G j ,H j l 1, L j l 1 ,l 2and M l1 ,l 2. Indeed, taking account of the symmetries, we enumerate:(3.36) ⎧#{□ j xx} = m, #{□ 0 xx} = 1,⎪⎨#{□ j } = xy l 1 m2 , #{□ 0 xy l 1} = m,⎪⎩ #{□ j y l 1y } = m2 (m + 1), #{□ 0 m(m + 1)l 2 y2l 1y l 2} = ,2whereas(3.37) ⎧⎨ #{G j } = m, #{H j l 1} = m 2 ,⎩ #{L j l 1 ,l 2} = m2 (m + 1)m(m + 1), #{M l1 ,l22} = .2Similarly as in Section 2, for j, l 1 , l 2 = 0, 1, . . ., m, let us introduce functionsΠ j l 1 ,l 2of (x, y 1 , . . .,y m ), symmetric with respect to the lower indices,33

34and let us seek necessary and sufficient conditions in order that there existsolutions (X, Y ) to the first auxiliary system defined precisely by:(3.38){ □0xx = Π 0 0,0, □ 0 xy l 1= Π 0 0,l 1, □ 0 y l 1y l 2= Π 0 l 1 ,l 2,□ j xx = Π j 0,0, □ j xy l 1 = Πj 0,l 1, □ j y l 1y l 2 = Πj l 1 ,l 2.3.39. Compatibility conditions for the first auxiliary system. As in Section2, the compatibility conditions for this system will simply be obtainedby computing the cross differentiations. The following statement generalizesLemma 2.31 and also provides a proof of it, in the case m = 1.Lemma 3.40. For all j, l 1 , l 2 , l 3 = 0, 1, . . ., m, we have the cross differentiationrelations(3.41)( ) ( ) m∑m∑□ j − □ j = − □ k y l 1y l 2y l 3 y l 1y l 3y l y l 1y l 2· □ j + □ k2y l 3y k y l 1y l 3· □ j .y l 2y kk=0Proof. To begin with, as a preliminary, let us generalize the Plücker identity(2.28). Let C 1 , C 2 , . . .,C m , D, E be (m + 2) column vectors in K m andintroduce the following notation for the m × (m + 2) matrix consisting ofthese vectors:(3.42) [C 1 |C 2 | · · · |C m |D|E].Extracting columns from this matrix, we shall construct m×m determinantswhich are modification of the following “fundamental” determinant(3.43) |C 1 | · · · |C m | ≡ ∣ ∣ ∣ ∣ C1 | · · · | j 1C j1 | · · · | j 2C j2 | · · · |C m∣ ∣ ∣ ∣ .Here and in the sequel, we use a double vertical line in the beginning and inthe end to denote a determinant. Also, we emphasize two distinct columns,the j 1 -th and the j 2 -th, where j 2 > j 1 , since we will modify them. Forinstance in this matrix, let us replace these two columns by the column Dand by the column E, which yields the determinant(3.44)∣ ∣ ∣ C1 | · · · | j 1D| · · · | j 2E| · · · |C m∣ ∣∣ ∣.In this notation, one should understand that only the j 1 -th and the j 2 -thcolumns are distinct from the columns of the fundamental m × m determinant(3.43). With this notation at hand, we can now formulate and prove apreliminary lemma that will be useful later.k=0

Lemma 3.45. The following quadratic identity between determinants holdstrue:(3.46)⎧∣ ∣ ∣ ∣ ∣ ⎪⎨ C1 | · · · | j 1D| · · · | j 2E| · · · |C n ∣ · ∣C1 | · · · | j 1C j1 | · · · | j 2C j2 | · · · |C n == ∣ ∣ ∣ ∣ ∣ C1 | · · · | j 1D| · · · | j 2C j2 | · · · |C n · ∣C1 | · · · | j 1C j1 | · · · | j 2E| · · · |C n ∣ −⎪⎩− ∣ ∣ ∣ ∣ C1 | · · · | j 1E| · · · | j 2C j 2| · · · |C n · ∣C1 | · · · | j 1C j1 | · · · | j 2D| · · · |C n .Proof. After some permutations of columns, this identity amounts to(3.47)⎧⎪⎨⎪⎩|C 1 | · · · |C m−2 |D|E | · |C 1 | · · · |C m−2 |C m−1 |C m | == |C 1 | · · · |C m−2 |D|C m | · |C 1 | · · · |C m−2 |C m−1 |E | −− ||C 1 | · · · |C m−2 |E|C m | · |C 1 | · · · |C m−2 |C m−1 |D |.To establish this identity, we introduce some notation. If A and B are verticalvectors in K m and if i 1 , i 2 = 1, . . .,m with i 1 < i 2 , we denote(3.48) ∆ 2 i 1 ,i 2(A|B) :=∣ A ∣i 1B i1 ∣∣∣.A i2 B i2If |A 1 |A 2 |A 3 | · · · |A m | is a m×m determinant, and if i 1 , i 2 = 1, . . .,m withi 1 < i 2 , we denote by M m−2i 1 ,i 2(A 3 | · · · |A m ) the (m−2)×(m−2) determinantobtained from the matrix [A 3 | · · · |A m ] by erasing the i 1 -th line and the i 2 -thline. Without proof, we recall an elementary classical formula(3.49)⎧⎨ |A 1 |A 2 |A 3 | · · · |A m | =⎩= ∑1i 1

36∑1i 1

often be denoted by the sign “·”, for clarity. Here is the computation:(3.53) ⎧ ( ) ( )□ j − □ j =y l 1y l 2y l 3 y l 1y l 3y l 2(= ∂(∆ y 0 | · · · | j y l 1y l 2| · · · |y m) ) (−∂(∆ y 0 | · · · | j y l 1y l 3| · · · |y m) )∂y l 3∆ (y 0 | · · · |y m ) ∂y l 2∆ (y 0 | · · · |y m )⎡∆ ( y 0 y l 3| · · · | j y l 1y l 2| · · · |y m) ⎤· ∆ + · · ·+⎪⎨ = 1+ ∆ ( y 0 | · · · | j y l 3y l 1y l 2| · · · |y m) · ∆ + · · ·+a [∆] 2 + ∆ ( y 0 | · · · | j y l 1y l 2| · · · |y l 3y m) · ∆−⎢⎣ − ∆ ( y 0 | · · · | j y l 1y l 2| · · · |y m) · [∆ ( −y 0 y l 3| · · · |y m) + · · ·+ ⎥+∆ ( y 0 | · · · |y l 3y m)] ⎦⎡∆ ( y 0 y l 2| · · · | j y l 1y l 3| · · · |y m) ⎤· ∆ + · · ·+− 1+ ∆ ( y 0 | · · · | j y l 2y l 1y l 3| · · · |y m) · ∆ + · · ·+a [∆] 2 + ∆ ( y 0 | · · · | j y l 1y l 3| · · · |y l 2y m) · ∆−⎢⎣ − ∆ ( y 0 | · · · | j y l 1y l 3| · · · |y m) · [∆ ( .y 0 y l 2| · · · |y m) + · · ·+ ⎥⎪⎩+∆ ( y 0 | · · · |y l 2y m)] ⎦37Crucially, we observe that all the determinants involving a third order derivativeupon one of their columns kill each other and disappear: we haveunderlined them with a appended. However, it still remains plenty of determinantsinvolving a second order derivative upon two different columns.We must transform all of them and express them in terms of determinantsinvolving a second order derivative upon only one column. To this aim, as anapplication of our preliminary Lemma 3.45, we have the following relations,valid for j 1 , j 2 , l 1 , l 2 , l 3 , l 4 = 0, . . ., m and j 1 < j 2 :(3.54)⎧∆ ( y ⎪⎨0 | · · · | j 1y l 1y l 2| · · · | j 2y l 3y l 4| · · · |y m) · ∆ ( y 0 | · · · | j 1y j 1| · · · | j 2y j 2| · · · |y m) == ∆ ( y 0 | · · · | j 1y l 1y l 2| · · · | j 2y j 2| · · · |y m) · ∆ ( y 0 | · · · | j 1y j 1| · · · | j 2y l 3y l 4| · · · |y m)⎪⎩− ∆ ( y 0 | · · · | j 1y l 3y l 4| · · · | j 2y j 2| · · · |y m) · ∆ ( y 0 | · · · | j 1y j 1| · · · | j 2y l 1y l 2| · · · |y m) .With these formulas, we may transform the lines number 3, 4, 5 and 8, 9, 10of (3.53). Also, we observe that the lines 6, 7 and 11, 12 of (3.53) involvedeterminants having a single second order derivative. Taking account of the1[∆] 2 factor, we deduce that the lines 6, 7 and 11, 12 of (3.53) may already beexpressed as sums of square functions. Achieving all these transformations,

38we may rewrite (3.53) as follows(3.55)⎧ ( ) ( )□ j − □ j =y l 1y l 2y l 3 y l 1y l 3y l 2⎡∆ ( y 0 y l 3| · · · | j y j | · · · |y m) · ∆ ( y 0 | · · · | j y l 1y l 2| · · · |y m) ⎤−= 1− ∆ ( y l 1y l 2| · · · | j y j | · · · |y m) · ∆ ( y 0 | · · · | j y 0 y l 3| · · · |y m) ++ · · ·+[∆] 2 ⎢⎣ + ∆ ( y 0 | · · · | j y l 1y l 2| · · · |y m) · ∆ ( −y 0 | · · · | j y j | · · · |y l 3y m) −⎥− ∆ ( y 0 | · · · | j y l 3y m | · · · |y m) · ∆ ( y 0 | · · · | j y j | · · · |y l 1y ) ⎦l 2⎪⎨m∑− □ j y l 1y l 2 □k y l 3y− k⎪⎩k=0⎡∆ ( y 0 y l 2| · · · | j y j | · · · |y m) · ∆ ( y 0 | · · · | j y l 1y l 3| · · · |y m) ⎤−− 1− ∆ ( y l 1y l 3| · · · | j y j | · · · |y m) · ∆ ( y 0 | · · · | j y 0 y l 2| · · · |y m) ++ · · ·+[∆] 2 ⎢⎣ + ∆ ( y 0 | · · · | j y l 1y l 3| · · · |y m) · ∆ ( +y 0 | · · · | j y j | · · · |y l 2y m) −⎥− ∆ ( y 0 | · · · | j y l 2y m | · · · |y m) · ∆ ( y 0 | · · · | j y j | · · · |y l 1y ) ⎦l 3m∑+ □ j y l 1y l 3 □k y l 2y. kk=0Notice that two pairs of “cdots” terms + · · ·+ appearing in the lines 3,4 and 8, 9 of (3.53) are replaced by a single “cdots” term + · · ·+ in thelines 4 and 9 of (3.55). Importantly, we point that in the “middle” of the two“cdots” terms + · · ·+ appearing in the lines 4 and 10 of (3.55) just above,there are two terms which do not occur: they simply correspond to the twounderlined terms having a appended appearing in the lines 4 and 9 (3.53).

1Now, taking account of the factor , we can re-express all the terms[∆] 2of (3.55) as sums of square functions:⎧ ( ) ( )□ j − □ j =y l 1y l 2 y l 1y l 3(3.56)⎪⎨⎪⎩=−−+y l 3m∑k=0; k≠jy l 2□ k y l 3y k □ j y l 1y l 2 −m∑□ j y l 1y l 2 □k y l 3y− kk=0m∑k=0; k≠j□ k y l 2y k □ j y l 1y l 3 +m∑□ j y l 1y l 3 □k y l 2y. kk=0m∑k=0; k≠jm∑k=0; k≠j□ k y l 1y l 2□ j y l 3y k −□ k y l 1y l 3□ j y l 2y k +Finally, we observe that in the two pairs of sums having k ≠ j appearingin the lines 2 and 4 just above, we can include the term k = j in each pair,because these two terms are immediately killed inside the correspondingpair. In conclusion, after a final obvious killing of four (among six) completesums in this modification of (3.56), we obtain the desired formula (3.41),with two sums. This completes the proof of Lemma 3.40 and also at thesame occasion, the proof of Lemma 2.31.3.57. Compatibility conditions for the first auxiliary system. Accordingto the (approximate) identities (3.3), taking account of the explicit definitions(3.30) of the square functions, we have{ □0 ∼ xx = Xxx , □ 0 ∼xy l 1 = Xxy l 1 , □ 0 ∼y l 1y l 2 = Xy l 1y l 2 ,(3.58)□ j xx ∼ = Y jxx, □ j xy l 1∼ = Yjxy l 1 , □j y l 1y l 2∼ = Yjy l 1y l 2 .Consequently, the first auxiliary system (3.38) looks approximatively like acomplete second order system of partial differential equations in the (m+1)independent variables (x, y) and in the (m+1) dependent variables (X, Y ).By means of elementary algebraic operations, one may transform this systemin a true second order complete system, solved with respect to the toporder derivatives, namely of the form{Xxx = Λ 0 0,0 , X xy l 1 = Λ 0 0,l 1, X y l 1y l 2 = Λ 0 l 1 ,l 2,(3.59)Y jxx = Λ j 0,0,Y jxy l 1 = Λj 0,l 1, Y y l 1y l 2 = Λ j l 1 ,l 2,where the Λ k 1j 1 ,j 2are local K-analytic functions of (x, y l 1, X, Y j , X x , X y l 1 , Y jx , Y jy l 1 ).For such a system, the compatibility conditions [which are necessary andsufficient for the existence of a solution (X, Y )] follow by obvious cross39

40differentiation. Coming back to the system 3.38, these compatibilityconditions amount to the quadratic-like compatibility conditions expressedin Lemma 3.40. In conclusion, we have proved the following intermediatestatement.Proposition 3.60. There exist functions X, Y j solving the first auxiliarysystem (3.38) of nonlinear second order partial differential equations if andonly if the right-hand side functions Π j l 1 , l 2(x, y) satisfy the quadratic compatibilityconditions(3.61)∂Π j l 1 ,l 2∂y l 3− ∂Πj l 1 ,l 3∂y l 2m∑m= − Π k l 1 ,l 2 · Π j l 3 ,k + ∑Π k l 1 ,l 3 · Π j l 2 ,k ,k=0k=0for j, l 1 , l 2 , l 3 = 0, 1, . . ., m.3.62. Principal unknowns. As there are (m + 1) more square (or Pi) functionsthan the functions G j , H j l 1, L j l 1 ,l 2and M l1 ,l 2defined by (3.35), wecannot invert directly the linear system (3.35) (which is of maximal rank).Hence we must choose (m+1) specific square functions, calling them principalunknowns, and similarly as in §2.34, the best choice is to choose □ 0 xxand □ j xx , for j = 1, . . .,m. For clarity, it will be useful to adopt the notationalequivalences(3.63) Θ 0 ≡ Π 0 0,0 and Θ j ≡ Π j j,j .Then we may quasi-inverse the system (3.35), which yields :(3.64) ⎧Π j 0,0 = □j xx = −Gj ,Π j 0,l 1= □ j xy⎪⎨= −1 l 12 Hj l 1+ 1 2 δj l 1Θ 0 ,Π j l 1 ,l 2= □ j y l 1y = l 2 −Lj l 1 ,l 2+ 1 2 δj l 1L l 2l2 ,l 2+ 1 2 δj l 2L l 1l1 ,l 1+ 1 2 δj l 1Θ l 2+ 1 2 δj l 2Θ l 1,Π 0 0,l 1= □ 0 xy l 1= 1 2 Ll 1l1 ,l 1+ 1 2 Θl 1,⎪⎩Π 0 l 1 ,l 2= □ 0 y l 1y = M l 2 l 1 ,l 2.Before replacing these new expressions of the functions Π j 0,0, Π j 0,l 1, Π j l 1 ,l 2,Π 0 l 1 ,l 2and Π l 10,l1into the compatibility conditions (3.61), it is necessary toexpound first (3.61), taking account of the original splitting of the indices inthe two sets {0} and {1, 2, . . ., m}. This yields six families of compatibility

conditions, totally equivalent to the compact identities (3.61):(3.65) ⎧( )Π j 0,0 −(Π j 0,l 1)x m = −Π0 0,0 Π j l 1 ,0 − ∑m Π k 0,0 Π j l 1 ,k + Π0 0,l 1Π j 0,0 + ∑Π k 0,l 1Π j 0,k ,⎪⎨⎪⎩y l 1( )(Π j l 1 ,l 2)x − Π j l 1 ,0y l 2(Π j l 1 ,l 2)y l 3)−(Π j l 1 ,l 3 y l 2k=1k=1m= −Π 0 l 1 ,l 2Π j 0,0 − ∑m Π k l 1 ,l 2Π j 0,k + Π0 l 1 ,0 Πj l 2 ,0 + ∑Π k l 1 ,0 Πj l 2 ,k ,k=141k=1m= −Π 0 l 1 ,l 2Π j l 3 ,0 − ∑m Π k l 1 ,l 2Π j l 3 ,k + Π0 l 1 ,l 3Π j l 2 ,0 + ∑Π k l 1 ,l 3Π j l 2 ,k ,k=1(Π00,0)y − ( Π 0 l 1 0,l 1)x = −Π0 0,0 Π0 l 1 ,0 − ∑m Π k 0,0a Π0 l 1 ,k + Π0 0,l 1Π 0 0,0 + ∑m Π k 0,la 1Π 0 0,k ,k=1( )Π0l1 ,l 2 x − ( Π 0 l 1 ,0)y m = l −Π0 2 l 1 ,l 2Π 0 0,0 − ∑m Π k l 1 ,l 2Π 0 0,k + Π0 l 1 ,0 Π0 l 2 ,0 + ∑Π k l 1 ,0 Π0 l 2 ,k ,k=1(Π0l1 ,l 2)y − ( Π 0 l 3 l 1 ,l 3)y m = l −Π0 2 l 1 ,l 2Π 0 l 3 ,0 − ∑m Π k l 1 ,l 2Π 0 l 3 ,k + Π0 l 1 ,l 3Π 0 l 2 ,0 + ∑Π k l 1 ,l 3Π 0 l 2 ,k .k=1k=1k=1k=1k=13.66. Convention about sums. Up to the end of Section 4, we shall abbreviateany sum ∑ mk=1 or ∑ mp=1 as ∑ k or ∑ p. Such sums will appear veryfrequently. For all other sums, we shall precisely write down the domain ofvariation of the summation index.3.67. Continuation. Thus, we have to replace (3.64) in the six identities(3.65). Firstly, let us expose all the intermediate steps in dealing with

42the first identity (3.65) 1 . Replacing plainly (3.64) in (3.65) 1 , we get:(3.68)⎧ ( ) )Π j 0,0 −(Π j 0,l = 1 x Gj y + 1 l 12 Hj l 1 ,x − 1 2 δj l 1Θ 0 x =y l 1= 1 2 Θ0 H j l 1− 1 2 δj l 1Θ 0 Θ 0 −− ∑ ( ( −G k) −L j l 1 ,k + 1 2 δj l 1L k k,k + 1 2 δj k Ll 1l1 ,l 1+ 1 2 δj l 1Θ k + 1 2 δj k)+ Θkk⎪⎨⎪⎩( 1+2 Ll 1l1 ,l 1+ 1 1) (−G2 Θl j ) ++ ∑ (− 1 2 Hk l 1+ 1 2 δk l 1Θ)(− 0 1 2 Hj k + 1 )2 δj k Θ0 =k= 1 2 Hj l 1Θ 0 − 1a 2 δj l 1Θ 0 Θ 0 − ∑ G k L j l 1 ,k + 1 ∑2 δj l 1kk+ 1 ∑2 δj l 1G k Θ k + 1 2 Gj Θ l 1k− 1 4 Hj l 1Θ 0 − 1a 4 Hj l 1Θ 0 + 1a 4 δj l 1Θ 0 Θ 0 .cG k L k k,k + 1 2 Gj L l 1l1 ,l 1b+− 1 2 Gj L l 1l1 ,l 1− 1b 2 Gj Θ l 1+ 1c 4∑Hl k 1H j k −kEliminating the underlined vanishing terms with the letters a, b, c and dappended, multiplying by −2 and reorganizing the identity so as to put theterm δ j l 1Θ 0 x solely in the left-hand side, we obtain the relation(3.69) ⎧δ⎪⎨j l 1Θ 0 x = −2 G j + y l 1 Hj l 1 ,x + 2 ∑ G k L j l 1 ,k − ∑δj l 1G k L k k,k−kk⎪⎩− 1 ∑Hl k 2 1H j k − ∑δj l 1G k Θ k + 1 2 δj l 1Θ 0 Θ 0 .k3.70. Conventions for simplifications of formal expressions. Before proceedingfurther, let us explain how we will organize the computations withthe formal expressions we shall encounter until the end of Section 4. Ourmain goal is to devise a methodology of writing formal computations whichenables to check every computation visually, without being forced to rebuildany intermediate step. In fact, it would be unsatisfactory to justclaim that Theorem 1.7 (3) follows by hidden massive formal computations,so that we have to guide the rigorous and demanding reader until the veryextremal branches of our coral tree of formal computations.As an example, suppose that we have to simplify the equation 0 = A x +B y +A−B +2C − 1 D − 2 A+ 1 D +E +B −2 C. In the beginning, the3 3 6terms A x and B y are differentiated once and they do not simplify with otherterms. To distinguish them, we underline them plainly and we copy the ninek

43remaining terms afterwards:(3.71) ⎧⎪⎨0 = A x + B y +⎪⎩+ A 1− B a+ 2C b− 1 3 D 2− 2 3 A 1+ 1 6 D 2+ E 3+ B a− 2C b.Here, each remaining term is also underlined, with a number or with a letterappended. For reasons of typographical readability, we never underline thesign, + or − of each term; however, it should be understood that every termalways includes its (not underlined) sign. Until the end of Section 4, we shalluse the roman alphabetic letters a, b, c, etc. inside an octagon to exhibitthe vanishing terms. As readily checked by the eyes, we indeed have −B a+B a= 0 and 2 C b−2 C b= 0. Also, until the end of Section 4, we shall usethe numbers 1, 2, 3, etc. inside a square □ to exhibit the remaining terms,collected in a certain order. The numbers have the following signification:after the simplifications, the equation (3.71) may be written⎧⎨ 0 = A x + B y +(3.72)⎩ + 1 3 A − 1 6 D + E.Here, the plainly underlined terms A x + B y do not count in the numbering(their number is zero, for instance) and the first term of the second line 1 A 3correspond to the addition of all terms 1 in (3.69). Analogously, the secondterm − 1 D correspond to the addition of all terms 2 in (3.69). Again, this6guiding facilitates the checking of the correctness of the computation, usingsimply the eyes. No hidden delicate computational step is “left to the reader”for the convenience of the writer.This principle will be constantly used until the end of Section 4; it hasbeen systematically used in [M2004] and it could be applied in various othercontexts. Again, the advantage is that it enables to check the correctness ofall the formal computations just by reading, without having to write anythingmore. This is also useful for the author.3.73. Choice of an ordering. Until the end of Section 4, we shall have todeal with terms G, H, L, M, Θ together with indices and partial derivativesup to order two. In order to organize the formal expressions in a waywhich provides an easier deciphering, it is convenient to introduce an orderbetween these differential monomials. In a symbolic index-free notation, wechoose:(3.74) G < H < L < M < Θ.It follows for instance that G < GH < GL < HHL < HLMΘ. Also, if asum appears, we choose: GM < ∑ GM.

44Here, we have only considered terms of order zero, without partial differentiation.The first order partial differentiations are (·) x and (·) y , again insymbolic notation, dropping the indices. We choose:(3.75)G x < G y < H x < H y < L x < L y < M x < M y < Θ x < Θ y < G < · · · .For second order derivatives, we choose:(3.76) G xx < G xy < G yy < H xx < · · · < Θ yy < G x < · · · .As a final general example including indices we have the inequalitiesm∑m∑(3.77) H j < l 1 ,y l 2 Lj l 1 ,l 2 ,x < Gj M l1 ,l 2< G k M l2 ,k < H l 2kHl k 2 ,l 2,extracted from (II) of Theorem 1.7 (3).In the sequel, we shall call• terms of order 0 monomials like G, H, L M, G H M;• terms of order 1 monomials like G x , G x M, L x Θ;• terms of order 2 monomials like G xy , L yy , M xx (our terms of order twowill always be linear),k=1according to the top order partial derivatives.3.78. A mean of checking intuitively the validity of partial differentialrelations. Before replacing (3.64) in the five remaining identities (3.65) 2 ,(3.65) 3 , (3.65) 4 , (3.65) 5 and (3.65) 6 , let us observe that if we assume thatj ≠ l 1 in (3.69), then all the terms involving Θ vanish, so that we obtain thenontrivial partial differential equations:(3.79) 0 = −2 G j + y l 1 Hj l 1 ,x + 2 ∑ G k L j l 1 ,k − 1 ∑Hl k 2 1H j k ,kkfor j ≠ l 1 . Here, we have underlined the first order terms plainly, in orderto distinguish them from the terms of order zero. These equations coincideswith (I) of Theorem 1.7 (3), again specialized with j ≠ l 1 . Importantly,we notice that the choice of indices j ≠ l 1 is possible only if m 2.Thus, we have derived a subpart of (I) as a necessary condition for the pointequivalence to Y j XX= 0, j = 1, . . .,m 2. These first order equationsshow at once that there is a strong difference with the case m = 1.How can we confirm (at least informally) that the functions G j , H j l 1andL j l 1 ,l 2given by (3.35) in terms of X and Y j do indeed satisfy these equationsfor j ≠ l 1 ? Dropping the zero order terms in (3.79) above, we obtain anapproximated equation(3.80) 0 ≡ −2 G j y l 1 + Hj l 1 ,x .k=1

Here, the sign ≡ precisely means: “modulo zero order terms”. We claimthat this approximated equation is a consequence of the existence of X, Y j .Indeed, according to the approximation (3.58), together with the definition(3.35) of the functions G j and H j l 1, we have45(3.81){ Gj= −□ j xx ∼ = −Y jH j l 1= −2 □ j xy l 1xx,∼= −2 Y jxy l 1 .Differentiation of the first line with respect to y l 1and of the second line withrespect to x yields:(3.82) G j y l 1∼ = −Yjxxy l 1and H j l 1 ,x ∼ = −2 Y jxy l 1x ,so that we indeed have 0 ≡ −2 G j + y l 1 Hj l 1 ,x, approximatively and modulothe derivatives of order 0, 1 and 2 of the functions X, Y j .Similar verifications have been effected constantly in our manuscript inorder to control the truth of the formal computations that we shall exposeuntil the end of Section 4.3.83. Continuation. From now on and up to the end of Section 4, the hardestcomputational core of the proof may — at last — be developed. Furtheramazing computational obstacles will be encountered.Replacing plainly (3.64) in (3.65) 2 , we get:(3.84)( )(Π j l 1 ,l 2)x − Π j l 1 ,0y l 2= −L j l 1 ,l 2 ,x + 1 2 δj l 1L l 2l2 ,l 2 ,x + 1 2 δj l 2L l 1l1 ,l 1 ,x + 1 2 δj l 1Θ l 2 x ++ 1 2 δj l 2Θ l 1 x + 1 2 Hj l 1 ,y l 2 − 1 2 δj l 1Θ 0 y l 2 == −Π 0 l 1 ,l 2 · Π j 0,0 − ∑ Π k l 1 ,l 2 · Π j 0,k + Π0 l 1 ,0 · Πj l 2 ,0 + ∑ Π k l 1 ,0 · Πj l 2 ,k =kk= M l1 ,l 2G j − ∑ (−L k l 1 ,l 2+ 1 2 δk l 1L l 2l2 ,l 2+ 1 2 δk l 2L l 1l1 ,l 1+ 1 2 δk l 1Θ l 2+ 1 )2 δk l 2Θ l 1·k·(− 1 2 Hj k + 1 )2 δj k Θ0 +( 1+2 Ll 1l1 ,l 1+ 1 )2 Θl 1·(− 1 2 Hj l 2+ 1 )2 δj l 2Θ 0 ++ ∑ (− 1 2 Hk l 1+ 1 )2 δk l 1Θ 0 ·k(· −L j l 2 ,k + 1 2 δj l 2L k k,k + 1 2 δj k Ll 2l2 ,l 2+ 1 2 δj l 2Θ k + 1 )2 δj k Θl 2.

46Developing the products and ordering each monomial, we get:(3.85)= G j M l1 ,l − 1 ∑2+ 1 1 24 Hj l 1L l 2l2 ,l 2+ 1a 4 Hj l 2L l 1l1 ,l 1b++ 1 4 Hj l 1Θ l 2ck+ 1 4 Hj l 2Θ l 1H j k Lk l 1 ,l 22d+ 1 2 Lj l 1 ,l 2Θ 0 e− 1 4 δj l 1L l 2l2 ,l 2Θ 0 f−− 1 4 δj l 2L l 1l1 ,l 1Θ 0 − 1g 4 δj l 1Θ 0 Θ l 2− 1h 4 δj l 2Θ 0 Θ l 1− 1i 4 Hj l 2L l 1l1 ,l 1b++ 1 4 δj l 2L l 1l1 ,l 1Θ 0 − 16 4 Hj l 2Θ l 1+ 1d 4 δj l 2Θ 0 Θ l 1+ 1 ∑Hl ki 2 1L j l 2 ,k− 1 4∑Hl k 1L k k,kk4− 1 4 Hj l 1L l 2l2 ,l 2− 1 ∑a 4 δj l 2Hl k 1Θ kk5k3− 1 4 Hj l 1Θ l 2− 1 2 Lj l 2 ,l 1Θ 0 + 1e 4 δj l 2L l 1l1 ,l 1Θ 0 + 1g 4 δj l 1L l 2l2 ,l 2Θ 0 + 1f 4 δj l 2Θ 0 Θ l 1+7+ 1 4 δj l 1Θ 0 Θ l 2.h−−cWe simplify according to our general principles and we reorganize the equalitybetween the first two lines of (3.84) and (3.85) so as to put all terms Θ xin the left-hand side of the equality and to put all remaining terms in theright-hand side, respecting the order of §3.73. We get:(3.86)12 δj l 1Θ l 2x + 1 2 δj l 2Θ l 1x − 1 2 δj l 1Θ 0 y l 2 == − 1 2 Hj l 1 ,y l 2 + Lj l 1 ,l 2 ,x − 1 2 δj l 1L l 2l2 ,l 2 ,x − 1 2 δj l 2L l 1l1 ,l 1 ,x ++ G j M l1 ,l 2− 1 ∑H j k2Lk l 1 ,l 2+ 1 ∑Hl k 2 1L j l 2 ,k −kk− 1 ∑4 δj l 2Hl k 1L k k,k − 1 ∑4 δj l 2Hl k 1Θ k + 1 4 δj l 2L l 1l1 ,l 1Θ 0 +kk+ 1 4 δj l 2Θ 0 Θ l 1.Here, we have underlined plainly the four first order terms appearing in thesecond line.

Next, replacing plainly (3.64) in (3.65) 3 , we get:(3.87) (Π j l 1 ,l 2)y l 3)−(Π j l 1 ,l 3 y l 2== −L j l 1 ,l 2 ,y l 3 + 1 2 δj l 1L l 2l 2 ,l 2 ,y l 3 + 1 2 δj l 2L l 1l 1 ,l 1 ,y l 3 + 1 2 δj l 1Θ l 2y l 3 + 1 2 δj l 2Θ l 1y l 3 ++ L j l 1 ,l 3 ,y l 2 − 1 2 δj l 1L l 3l 3 ,l 3 ,y l 2 − 1 2 δj l 3L l 1l 1 ,l 1 ,y l 2 − 1 2 δj l 1Θ l 3y l 2 − 1 2 δj l 3Θ l 1y l 2 == −Π 0 l 1 ,l 2 · Π j l 3 ,0 − ∑ Π k l 1 ,l 2 · Π j l 3 ,k + Π0 l 1 ,l 3 · Π j l 2 ,0 + ∑kk= −M l1 ,l 2 ·(− 1 2 Hj l 3+ 1 )2 δj l 3Θ 0 − ∑ k)+ 1 2 δk l 2L l 1l1 ,l 1+ 1 2 δk l 1Θ l 2+ 1 2 δk l 2Θ l 1+ 1 2 δj k Ll 3l3 ,l 3+ 1 2 δj l 3Θ k + 1 2 δj k Θl 3+ M l1 ,l 3 ·)+(− 1 2 Hj l 2+ 1 )2 δj l 2Θ 0 + ∑ k)+ 1 2 δk l 3L l 1l1 ,l 1+ 1 2 δk l 1Θ l 3+ 1 2 δk l 3Θ l 1+ 1 2 δj k Ll 2l2 ,l 2+ 1 2 δj l 2Θ k + 1 2 δj k Θl 2).Π k l 1 ,l 3 · Π j l 2 ,k =(−L k l 1 ,l 2+ 1 2 δk l 1L l 2l2 ,l 2+(· −L j l 3 ,k + 1 2 δj l 3L k k,k +(−L k l 1 ,l 3+ 1 2 δk l 1L l 3l3 ,l 3+(· −L j l 2 ,k + 1 2 δj l 2L k k,k +Developing the products and ordering each monomial, we get:(3.88)= 1 2 Hj l 3M l1 ,l 2− 11 2 δj l 3M l1 ,l 2Θ 0 − ∑ L k l 1 ,l 2L j l 3 ,k+ 1 ∑162 δj l 3k+ 1 2 Lj l 1 ,l 2L l 3l3 ,l 3+ 1 ∑a 2 δj l 3L k l 1 ,l 2Θ kk136+ 1 2 Lj l 1 ,l 2Θ l 3bk47L k l 1 ,l 2L k k,k+ 1 2 Ll 2l2 ,l 2L j l 3 ,l 1c−− 1 4 δj l 3L l 2l2 ,l 2L l 1l1 ,l 1− 14 4 δj l 1L l 2l2 ,l 2L l 3l3 ,l 3− 1d 4 δj l 3L l 2l2 ,l 2Θ l 1− 1e 4 δj l 1L l 2l2 ,l 2Θ l 3+f+ 1 2 Ll 1l1 ,l 1L j l 3 ,l 2− 1g 4 δj l 3L l 1l1 ,l 1L l 2l2 ,l 2− 1h 4 δj l 2L l 1l1 ,l 1L l 3l3 ,l 3− 1i 4 δj l 3L l 1l1 ,l 1Θ l 2−j7+− 1 4 δj l 2L l 1l1 ,l 1Θ l 3k+ 1 2 Lj l 3 ,l 1Θ l 2l− 1 4 δj l 3L l 1l1 ,l 1Θ l 210− 1 4 δj l 1L l 3l3 ,l 3Θ l 2−m− 1 4 δj l 3Θ l 2Θ l 1n− 1 4 δj l 1Θ l 2Θ l 3o+ 1 2 Lj l 3 ,l 2Θ l 1p− 1 4 δj l 3L l 2l2 ,l 2Θ l 112−− 1 4 δj l 2L l 3l3 ,l 3Θ l 1q− 1 4 δj l 3Θ l 1Θ l 218− 1 4 δj l 2Θ l 1Θ l 3−r

48− 1 2 Hj l 2M l1 ,l 32+ 1 2 δj l 2M l1 ,l 3Θ 0 15− 1 2 Lj l 1 ,l 3L l 2l2 ,l 2c− 1 2 δj l 2∑k+ ∑ kL k l 1 ,l 3Θ k 14L k l 1 ,l 3L j l 2 ,k− 1 2 Lj l 1 ,l 3Θ l 25l− 1 ∑2 δj l 2L k l 1 ,l 3L k k,kk− 1 2 Ll 3l3 ,l 3L j l 2 ,l 1a+8−+ 1 4 δj l 2L l 3l3 ,l 3L l 1l1 ,l 13+ 1 4 δj l 1L l 3l3 ,l 3L l 2l2 ,l 2d+ 1 4 δj l 2L l 3l3 ,l 3Θ l 1q+ 1 4 δj l 1L l 3l3 ,l 3Θ l 2− 1 2 Ll 1l1 ,l 1L j l 2 ,l 3g+ 1 4 δj l 2L l 1l1 ,l 1L l 3l3 ,l 3i+ 1 4 δj l 3L l 1l1 ,l 1L l 2l2 ,l 2h+ 1 4 δj l 2L l 1l1 ,l 1Θ l 3+ 1 4 δj l 3L l 1l1 ,l 1Θ l 2j− 1 2 Lj l 2 ,l 1Θ l 3b+ 1 4 δj l 2L l 1l1 ,l 1Θ l 3k+ 1 4 δj l 1L l 2l2 ,l 2Θ l 3+f9−m++ 1 4 δj l 2Θ l 3Θ l 117+ 1 4 δj l 1Θ l 3Θ l 2o− 1 2 Lj l 2 ,l 3Θ l 1p+ 1 4 δj l 2L l 3l3 ,l 3Θ l 111++ 1 4 δj l 3L l 2l2 ,l 2Θ l 1e+ 1 4 δj l 2Θ l 1Θ l 3r+ 1 4 δj l 3Θ l 1Θ l 2.nWe simplify and we reorganize the equality between the second and thirdlines of (3.88) and (3.85) so as to put all terms Θ y in the left-hand side ofthe equality and to put all remaining terms in the right-hand side, respecting

49the order of §3.73. We get:(3.89)12 δj l 1Θ l 2− 1 y l 32 δj l 1Θ l 3+ 1 y l 22 δj l 2Θ l 1− 1 y l 32 δj l 3Θ l 1= y l 2= L j l 1 ,l 2 ,y l 3 − Lj l 1 ,l 3 ,y l 2 + 1 2 δj l 1L l 3l 3 ,l 3 ,y l 2 − 1 2 δj l 1L l 2l 2 ,l 2 ,y l 3 ++ 1 2 δj l 3L l 1l 1 ,l 1 ,y l 2 − 1 2 δj l 2L l 1l 1 ,l 1 ,y l 3 ++ 1 2 Hj l 3M l1 ,l 2− 1 2 Hj l 2M l1 ,l 3+ 1 4 δj l 2L l 3l3 ,l 3L l 1l1 ,l 1− 1 4 δj l 3L l 2l2 ,l 2L l 1l1 ,l 1++ ∑ kL k l 1 ,l 3L j l 2 ,k − ∑ kL k l 1 ,l 2L j l 3 ,k ++ 1 ∑2 δj l 3L k l 1 ,l 2L k k,k − 1 ∑2 δj l 2L k l 1 ,l 3L k k,k +kk+ 1 4 δj l 2L l 1l1 ,l 1Θ l 3− 1 4 δj l 3L l 1l1 ,l 1Θ l 2+ 1 4 δj l 2L l 3l3 ,l 3Θ l 1− 1 4 δj l 3L l 2l2 ,l 2Θ l 1++ 1 ∑2 δj l 3L k l 1 ,l 2Θ k − 1 ∑2 δj l 2L k l 1 ,l 3Θ k +kk+ 1 2 δj l 2M l1 ,l 3Θ 0 − 1 2 δj l 3M l1 ,l 2Θ 0 + 1 4 δj l 2Θ l 3Θ l 1− 1 4 δj l 3Θ l 2Θ l 1.Next, replacing plainly (3.64) in (3.65) 4 , we get:(3.90) (Π00,0)y l 1 − ( Π 0 0,l 1)x == Θ 0 y l 1 − 1 2 Ll 1l1 ,l 1 ,x − 1 2 Θl 1 x == −Π 0 0,0 · Π0 l 1 ,0 − ∑ Π k 0,0 · a Π0 l 1 ,k + Π0 0,l 1 · Π 0 0,0 + ∑ akk= − ∑ (−G k) · (M l1 ,k) + ∑ (− 1 2 Hk l 1+ 1 )2 δk l 1Θ 0 ·kk( 1·2 Lk k,k + 1 )2 Θk == ∑ kG k M l1 ,k − 1 ∑Hl k 4 1L k k,k − 1 ∑4kkΠ k 0,l 1 · Π 0 0,k =H k l 1Θ k + 1 4 Ll 1l1 ,l 1Θ 0 + 1 4 Θ0 Θ l 1.

50Reorganizing the equality so as to put the terms Θ x and Θ y alone in theleft-hand side, we get:(3.91)− 1 2 Θl 1x + Θ 0 y l 1= 1 2 Ll 1l1 ,l 1 ,x ++ ∑ kG k M l1 ,k − 1 ∑Hl k 4 1L k k,k − 1 ∑Hl k 4 1Θ k ++ 1 4 Ll 1l1 ,l 1Θ 0 + 1 4 Θ0 Θ l 1.kkNext, replacing plainly (3.64) in (3.65) 5 , we get:(3.92) ( )Π0l1 ,l 2 x − ( Π 0 )l 1 ,0y l 2 == M l1 ,l 2 ,x − 1 2 Ll 1l 1 ,l 1 ,y l 2 − 1 2 Θl 1y l 2 == −Π l1 ,l 2 · Π 0 0,0 − ∑ Π k l 1 ,l 2 · Π 0 0,k + Π0 l 1 ,0 · Π0 l 2 ,0 + ∑ Π k l 1 ,0 · Π0 l 2 ,k =kk= −M l1 ,l 2Θ 0 − ∑ (−L k l 1 ,l 2+ 1 2 δk l 1L l 2l2 ,l 2+ 1 2 δk l 2L l 1l1 ,l 1+k+ 1 2 δk l 1Θ l 2+ 1 ) (2 δk l 2Θ l 1 1·2 Lk k,k + 1 )2 Θk +))++( 12 Ll 1l1 ,l 1+ 1 2 Θl 1+ ∑ k·( 12 Ll 2l2 ,l 2+ 1 2 Θl 2(− 1 2 Hk l 1+ 1 2 δk l 1Θ 0 )· M l2 ,k == −M l1 ,l 2Θ 0 7 + 1 2∑L k l 1 ,l 2L k k,kk3+ 1 2∑L k l 1 ,l 2Θ kk6− 1 4 Ll 2l2 ,l 2L l 1l1 ,l 1 2−− 1 4 Ll 2l2 ,l 2Θ l 1− 1 4 Θl 1Θ l 2d+ 1 4 Ll 1l1 ,l 1Θ l 2a− 1 4 Ll 1l1 ,l 1L l 2l2 ,l 2− 1b 4 Ll 1l1 ,l 1Θ l 2− 1c 4 Ll 1l1 ,l 1Θ l 2−4− 1 4 Ll 2l2 ,l 2Θ l 1c+ 1 2 M l 2 ,l 1Θ 0 .75+ 1 4 Ll 2l2 ,l 2Θ l 1a− 1 4 Θl 1Θ l 2+ 1 4 Θl 1Θ l 28+ 1 4 Ll 1l1 ,l 1L l 2l2 ,l 2b+d− 1 2∑Hl k 1M l2 ,kk1+

51Multiplying by −2 and reorganizing the equality, we get:Θ l 1y l 2 = −Ll 1l 1 ,l 1 ,y l 2 + 2 M l 1 ,l 2 ,x+(3.93)+ ∑ kH k l 1M l2 ,k + 1 2 Ll 1l1 ,l 1L l 2l2 ,l 2− ∑ k+ 1 2 Ll 1l1 ,l 1Θ l 2+ 1 2 Ll 2l2 ,l 2Θ l 1− ∑ kL k l 1 ,l 2L k k,k +L k l 1 ,l 2Θ k ++ M l1 ,l 2Θ 0 + 1 2 Θl 1Θ l 2.Next, replacing plainly (3.64) in (3.65) 4 , we get:(3.94) (Π0l1 ,l 2)y l 3 − ( Π 0 l 1 ,l 3)y l 2 == M l1 ,l 2 ,y l 3 − M l1 ,l 3 ,y l 2 == −Π 0 l 1 ,l 2 · Π 0 l 3 ,0 − ∑ k( 1= −M l1 ,l 22 Ll 3l3 ,l 3+ 1 )2 Θl 3− ∑ k+ 1 2 δk l 2L l 1l1 ,l 1+ 1 2 δk l 1Θ l 2+ 1 2 δk l 2Θ l 1+ M l1 ,l 3( 12 Ll 2l2 ,l 2+ 1 2 Θl 2Π k l 1 ,l 2 · Π 0 l 3 ,k + Π0 l 1 ,l 3 · Π 0 l 2 ,0 + ∑ k)+ ∑ k+ 1 2 δk l 3L l 1l1 ,l 1+ 1 2 δk l 1Θ l 3+ 1 2 δk l 3Θ l 1M l3 ,k)+M l2 ,k).Π k l 1 ,l 3 · Π 0 l 2 ,k =(−L k l 1 ,l 2+ 1 2 δk l 1L l 2l2 ,l 2+(−L k l 1 ,l 3+ 1 2 δk l 1L l 3l3 ,l 3+Developing the products and ordering each monomial, we get:(3.95)= − 1 2 Ll 3l3 ,l 3M l1 ,l 2− 1a 2 M l 1 ,l 2Θ l 3+ ∑b kL k l 1 ,l 2M l3 ,k1− 1 2 Ll 2l2 ,l 2M l3 ,l 1c−− 1 2 Ll 1l1 ,l 1M l3 ,l 2− 1d 2 M l 3 ,l 1Θ l 2− 1e 2 M l 3 ,l 2Θ l 1+ 1f 2 Ll 2l2 ,l 2M l3 ,l 1c++ 1 2 M l 1 ,l 3Θ l 2e− ∑ kL k l 1 ,l 3M l2 ,k2+ 1 2 Ll 3l3 ,l 3M l2 ,l 1+ 1a 2 Ll 1l1 ,l 1M l2 ,l 3d++ 1 2 M l 2 ,l 1Θ l 3b+ 1 2 M l 2 ,l 3Θ l 1.fSimplifying, we obtain the family (IV) in the statement of Theorem 1.7 (3):(3.96) 0 = M l1 ,l 2 ,y l 3 − M l1 ,l 3 ,y l 2 − ∑ kL k l 1 ,l 2M l3 ,k + ∑ kL k l 1 ,l 3M l2 ,k.

523.97. Solving Θ 0 x , Θ0 , y l 1 Θl 1 x and Θ l 1. It is now easy to solve all first orderpartial derivatives of the functions Θ 0 and Θ l . Equation (3.93) alreadyy l 2provides the solution for Θ l 1. We state the result as an independent proposition.y l 2Proposition 3.98. As a consequence of the six families of equations (3.69),(3.86), (3.89), (3.91), (3.93) and (3.96) the first order derivatives Θ 0 x , Θ0 y l 1 ,Θ l 1 x and Θ l 1y l 2of the principal unknowns are given by:(3.99)⎧⎪⎨Θ 0 x = −2 G l 1y l 1 + Hl 1l1 ,x ++ 2 ∑ kG k L l 1l1 ,k − ∑ kG k L k k,k − 1 2∑kH k l 1H l 1k−⎪⎩− ∑ kG k Θ k + 1 2 Θ0 Θ 0 .(3.100)⎧Θ 0 y = 2 l 13 Ll 1l1 ,l 1 ,x − 1 3 Hl 1+ l 1 ,y l 1⎪⎨⎪⎩+ 2 3 Gl 1M l1 ,l 1+ 4 ∑G k M l1 ,k − 1 ∑H l 133 kL k l 1 ,l 1+kk+ 1 ∑Hl k 3 1L l 1l1 ,k − 1 ∑Hl k 2 1L k k,k − 1 ∑Hl k 2 1Θ k +k+ 1 2 Ll 1l1 ,l 1Θ 0 + 1 2 Θ0 Θ l 1.kk(3.101)⎧Θ l 1x = − 2 3 Hl 1+ 1l 1 ,y l 13 Ll 1l1 ,l 1 ,x +⎪⎨+ 4 3 Gl 1M l1 ,l 1+ 2 ∑G k M l1 ,k − 2 ∑H l 133 kL k l 1 ,l 1+kk+ 2 ∑Hl k 3 1L l 1l1 ,k − 1 ∑Hl k 2 1L k k,k − 1 ∑Hl k 2 1Θ k +⎪⎩k+ 1 2 Ll 1l1 ,l 1Θ 0 + 1 2 Θ0 Θ l 1.kk

⎧Θ l 1y l 2 = −Ll 1l 1 ,l 1 ,y l 2 + 2 M l 1 ,l 2 ,x+53(3.102)⎪⎨⎪⎩+ ∑ kH k l 1M l2 ,k + 1 2 Ll 1l1 ,l 1L l 2l2 ,l 2− ∑ k+ 1 2 Ll 1l1 ,l 1Θ l 2+ 1 2 Ll 2l2 ,l 2Θ l 1− ∑ k+ M l1 ,l 2Θ 0 + 1 2 Θl 1Θ l 2.L k l 1 ,l 2L k k,k +L k l 1 ,l 2Θ k +We notice that the right-hand side of (3.99) should be independent of l 1 ;this phenomenon will be explained in a while.Proof. For Θ 0 x in (3.99), it suffices to put j := l 1 in (3.69).To obtain Θ 0 y l 1 , we put j := l 2 and l 2 := l 1 in (3.86), which yields:(3.103)Θ l 1 x − 1 2 Θ0 y l 1 = −1 2 Hl 1l 1 ,y l 1 ++ G l 1M l1 ,l 1− 1 2+ 1 2− 1 4∑k∑k∑kH k l 1L l 1l1 ,k − 1 4H l 1kL k l 1 ,l 1+∑Hl k 1L k k,k −kH k l 1Θ k + 1 4 Ll 1l1 ,l 1Θ 0 + 1 4 Θ0 Θ l 1.We may easily solve Θ 0 y l 1 and Θl 1 x thanks to this equation (3.103) and thanksto (3.91): indeed, to obtain (3.100), it suffices to compute 4 3 · (3.91) + 2 3 ·(3.103); to obtain (3.101), it suffices to compute 2 3 · (3.91) + 4 3 · (3.103).Finally, (3.102) is a copy of (3.93). This completes the proof.3.104. Appearance of the crucial four families of first order partial differentialrelations (I), (II), (III) and (IV) of Theorem 1.7 (3). However,in solving Θ 0 x, Θ 0 , y l 1 Θl 1 x and Θ l 1from our six families of equations (3.69),y l 2(3.86), (3.89), (3.91), (3.93) and (3.96), only a subpart of these equations hasbeen used. We notice that the two families of equations (3.91) and (3.93)have been used completely and that the family of equations (3.96), whichdoes not involve Θ, coincides precisely with the system (IV) of Theorem1.7 (3). To insure that Θ 0 x , Θ0 , y l 1 Θl 1 x and Θ l 1as written in Proposi-y l 2tion 3.98 are true solutions, it is necessary and sufficient that they satisfythe remaining equations. Thus, we have to replace these solutions (3.99),(3.100), (3.101) and (3.102) in the three remaining families (3.69), (3.86)and (3.89).Firstly, let us insert inside (3.69) the value of Θ 0 x given by the equation(3.99), in which the index l 1 is replaced in advance by an arbitrary

54index l 2 . We get:(3.105)0 = −2G j y + l 1 2δj l 1G l 2y + l 2 Hj l 1 ,x − δj l 1H l 2l2 ,x ++ 2 ∑ G k L j l 1 ,k − ∑2δj l 1G k L l 2l2 ,k − ∑δj l 1G k L k k,k +kkka∑+ δ j l 1G k L k k,k − 1 ∑Hl k 2 1H j k + 1 ∑2 δj l 1Hl k 2H l 2k−kak∑ ∑− δ j l 1G k Θ k + δ j l 1G k Θ k + 1 2 δj l 1Θ 0 Θ 0 − 1kbkc 2 δj l 1Θ 0 Θ 0 .cbWe simplify, which yields the family (I) of partial differential relations ofTheorem 1.7 (3):k(3.106)0 = − 2 G j + 2 y l 1 δj l 1G l 2+ y l 2 Hj l 1 ,x − δj l 1H l 2l2 ,x +G k L j l 1 ,k − 2 ∑δj l 1G k L l 2l2 ,k ++ 2 ∑ kk− 1 ∑Hl k 2 1H j k + 1 ∑2 δj l 1kkH k l 2H l 2k.Secondly, let us insert inside (3.86) the values of Θ l 1 x , Θ l 2 x given by (3.101)and the value of Θ 0 given by (3.100). We place all the terms in the righthandside of the equality and we place the first order terms in the beginningy l 1(first three lines just below). We obtain:(3.107)0 = − 1 2 Hj l 1 ,y l 2 1+ L j l 1 ,l 2 ,x 4− 1 2 δj l 1L l 2l2 ,l 2 ,x5− 1 2 δj l 1L l 1l1 ,l 1 ,x6++ 1 3 δj l 1H l 2l 2 ,y l 2 2− 1 6 δj l 1L l 2l2 ,l 2 ,x5+ 1 3 δj l 2H l 1l 1 ,y l 1 3− 1 6 δj l 2L l 1l1 ,l 1 ,x6++ 1 3 δj l 1L l 2l2 ,l 2 ,x5− 1 6 δj l 1H l 2l 2 ,y l 2 2++ G j M l1 ,l 2 7 − 1 2− 1 4 δj l 2∑k− 2 3 δj l 1G l 2M l2 ,l 28∑kH j k Lk l 1 ,l 212+ 1 2∑Hl k 1L j l 2 ,kHl k 1Θ k + 1 4 δj l 2L l 1l1 ,l 1Θ 0 + 1c 4 δj l 2Θ 0 Θ l 1−db− 1 ∑3 δj l 1G k M l2 ,kkk1013+ 1 3 δj l 1∑k− 1 ∑4 δj l 2Hl k 1L k k,kkH l 2kL k l 2 ,l 214a−− 1 3 δj l 1∑kH k l 2L l 2l2 ,k15+

+ 1 ∑4 δj l 1Hl k 2L k k,kk− 2 3 δj l 2G l 1M l1 ,l 19+ 1 ∑4 δj l 2Hl k 1L k k,kke+ 1 4 δj l 1∑− 1 ∑3 δj l 2G k M l1 ,kakk+ 1 4 δj l 2∑+ 1 3 δj l 1G l 2M l2 ,l 2+ 2 ∑8 3 δj l 1G k M l2 ,k− 1 ∑4 δj l 1Hl k 2L k k,kkekk− 1 4 δj l 1∑k55Hl k 2Θ k − 1 4 δj l 1L l 2l2 ,l 2Θ 0 − 1g 4 δj l 1Θ 0 Θ l 2−hf11+ 1 3 δj l 2∑kH l 1kL k l 1 ,l 116− 1 3 δj l 2∑Hl k 1Θ k − 1 4 δj l 2L l 1l1 ,l 1Θ 0 − 1c 4 δj l 2Θ 0 Θ 0 d+b10− 1 6 δj l 1∑kH l 2kL k l 2 ,l 214k+ 1 6 δj l 1∑Hl k 2Θ k + 1 4 δj l 1L l 2l2 ,l 2Θ 0 + 1g 4 δj l 1Θ 0 Θ l 2.hfkH k l 1L l 1l1 ,kH k l 2L l 2l2 ,k1715+−Simplifying and ordering, we obtain the family (II) of partial differentialrelations of Theorem 1.7 (3):(3.108)0 = − 1 2 Hj l 1 ,y l 2 + 1 6 δj l 1H l 2l 2 ,y l 2 + 1 3 δj l 2H l 1l 1 ,y l 1 ++ L j l 1 ,l 2 ,x − 1 3 δj l 1L l 2l2 ,l 2 ,x − 2 3 δj l 2L l 1l1 ,l 1 ,x ++ G j M l1 ,l 2− 1 3 δj l 1G l 2M l2 ,l 2− 2 3 δj l 2G l 1M l1 ,l 1+ 1 ∑3 δj l 1G k M l2 ,k−− 1 ∑3 δj l 2G k M l1 ,k − 1 ∑H j k2Lk l 1 ,l 2+ 1 ∑Hl k 2 1L j l 2 ,k +k+ 1 6 δj l 1∑k+ 1 3 δj l 2∑kH l 2kL k l 2 ,l 2− 1 6 δj l 1∑H l 1kL k l 1 ,l 1− 1 3 δj l 2∑kkkH k l 2L l 2l2 ,k +H k l 1L l 1l1 ,k .kkThirdly, let us insert inside (3.89) the values of Θ l 2, of y l 3 Θl 3, of y l 2 Θl 1and y l 3of Θ l 1given by (3.102). We place all the terms in the right-hand side of they l 2equality and we place the first order terms in the beginning (first four lines

56just below). We obtain:(3.109)0 = 1 2 δj l 1L l 2− δ j l 2 ,l 2 ,y l 3 l 1M l2 ,l 3 ,x + 1a b 2 δj l 2L l 1− δ j l 1 ,l 1 ,y l 3 l 2M l1 ,l 3 ,x −c4− 1 2 δj l 1L l 3+ δ j l 3 ,l 3 ,y l 2 l 1M l3 ,l 2 ,x − 1d b 3 δj l 3L l 1+ δ j l 1 ,l 1 ,y l 2 l 3M l1 ,l 2 ,x +e3+ L j − L j + 1 l 1 ,l 2 ,y l 3 l 1 1 ,l 3 ,y l 2 2 2 δj l 1L l 3− 1 l 3 ,l 3 ,y l 2d 2 δj l 1L l 2+l 2 ,l 2 ,y l 3a+ 1 3 δj l 3L l 1− 1 l 1 ,l 1 ,y l 2e 2 δj l 2L l 1+l 1 ,l 1 ,y l 3c+ 1 2 Hj l 3M l1 ,l 2− 15 2 Hj l 2M l1 ,l 3+ 16 4 δj l 2L l 3l3 ,l 3L l 1l1 ,l 1− 1f 4 δj l 3L l 2l2 ,l 2L l 1l1 ,l 1g++ ∑ kL k l 1 ,l 3L j l 2 ,k− ∑ kL k l 1 ,l 2L j l 3 ,k+11+ 1 ∑2 δj l 3L k l 1 ,l 2L k k,kk+ 1 4 δj l 2L l 1l1 ,l 1Θ l 3jh12− 1 ∑2 δj l 2L k l 1 ,l 3L k k,kk− 1 4 δj l 3L l 1l1 ,l 1Θ l 2ki++ 1 4 δj l 2L l 3l3 ,l 3Θ l 1+ 1 ∑2 δj l 3L k l 1 ,l 2Θ k − 1 ∑2 δj l 2L k l 1 ,l 3Θ kknko++ 1 2 δj l 2M l1 ,l 3Θ 0 − 1p 2 δj l 3M l1 ,l 2Θ 0 + 1q 4 δj l 2Θ l 3Θ l 1− 1 ∑2 δj l 1Hl k 2M l3 ,kk− 1 4 δj l 1L l 2l2 ,l 2Θ l 3v8rl− 1 4 δj l 3L l 2l2 ,l 2Θ l 1− 1 4 δj l 3Θ l 2Θ l 1−s− 1 4 δj l 1L l 2l2 ,l 2L l 3l3 ,l 3+ 1 ∑t 2 δj l 1L l2 ,l 3L k k,k −ku+ 1 ∑2 δj l 1− 1 4 δj l 1L l 3l3 ,l 3Θ l 2− 1 2 δj l 1M l2 ,l 3Θ 0 y− 1 4 δj l 1Θ l 2Θ l 3− 1 ∑2 δj l 2Hl k 1M l3 ,kk− 1 4 δj l 2L l 1l1 ,l 1Θ l 3j10w−zkL k l 2 ,l 3Θ k x−− 1 4 δj l 2L l 1l1 ,l 1L l 3l3 ,l 3+ 1 ∑f 2 δj l 2L k l 1 ,l 3L k k,k −ki+ 1 ∑2 δj l 2L k l 1 ,l 3Θ k o−− 1 4 δj l 2L l 3l3 ,l 3Θ l 1− 1 2 δj l 2M l1 ,l 3Θ 0 p− 1 4 δj l 2Θ l 3Θ l 1l+rk+m

+ 1 ∑2 δj l 1Hl k 3M l2 ,kk+ 1 4 δj l 1L l 3l3 ,l 3Θ l 2w7+ 1 4 δj l 1L l 3l3 ,l 3L l 2l2 ,l 2− 1 ∑t 2 δj l 1L l3 ,l 2L k k,k +ku− 1 ∑2 δj l 1L k l 3 ,l 2Θ k x++ 1 4 δj l 1L l 2l2 ,l 2Θ l 3+ 1 2 δj l 1M l3 ,l 2Θ 0 y+ 1 4 δj l 1Θ l 3Θ l 2+ 1 ∑2 δj l 3Hl k 1M l2 ,kk+ 1 4 δj l 3L l 1l1 ,l 1Θ l 2k9v+zk+ 1 4 δj l 3L l 1l1 ,l 1L l 2l2 ,l 2− 1 ∑g 2 δj l 3L k l 1 ,l 2L k k,k +kh− 1 ∑2 δj l 3L k l 1 ,l 2Θ k n++ 1 4 δj l 3L l 2l2 ,l 2Θ l 1+ 1 2 δj l 3M l1 ,l 2Θ 0 q+ 1 4 δj l 3Θ l 1Θ l 2Simplifying and ordering, we obtain the family (III) of partial differentialrelations of Theorem 1.7 (3):sm.0 = L j l 1 ,l 2 ,y l 3 − Lj l 1 ,l 3 ,y l 2 + δj l 3M l1 ,l 2 ,x − δ j l 2M l1 ,l 3 ,x+k57(3.110)+ 1 2 Hj l 3M l1 ,l 2− 1 2 Hj l 2M l1 ,l 3++ 1 ∑2 δj l 1Hl k 3M l2 ,k − 1 ∑2 δj l 1Hl k 2M l3 ,k+k+ 1 ∑2 δj l 3Hl k 1M l2 ,k − 1 ∑2 δj l 2Hl k 1M l3 ,k+kk+ ∑ L k l 1 ,l 3L j l 2 ,k − ∑ L k l 1 ,l 2L j l 3 ,k .kkk3.111. Arguments for the proof of Theorem 1.7 (3): necessity and sufficiencyof (I), (II), (III), (IV). Let us summarize the implications that havebeen established so far, from the beginning of Section 3. Recall that m 2.• There exist functions X, Y j of (x, y) transforming the system y j xx =F j (x, y, y x ), j = 1, . . ., m, to the free particle system Y j XX = 0, j =1, . . ., m.⇓• There exist functions Π j l 1 ,l 2of (x, y), 0 j, l 1 , l 2 m, satisfying thefirst auxiliary system (3.38) of partial differential equations.⇓• There exist (principal unknowns) functions Θ 0 , Θ j satisfying the sixfamilies of partial differential equations (3.69), (3.86), (3.89), (3.91),(3.93) and (3.96).

58⇓• The functions G j , H j l 1, L j l 1 ,l 2and M l1 ,l 2satisfy the four families ofpartial differential equations (I), (II), (III) and (IV) of Theorem 1.7 (3).The four families of first order partial differential equations (3.99),(3.100), (3.101) and (3.102) satisfied by the principal unknowns will becalled the second auxiliary system. It is a complete system.To achieve the proof of Theorem 1.7 (3), we have to establish the reverseimplications. More precisely:• Some given functions G j , H j l 1, L j l 1 ,l 2= L j l 2 ,l 1and M l1 ,l 2= M l2 ,l 1of(x, y) satisfy the four families of partial differential equations (I), (II),(III) and (IV) of Theorem 1.7 (3), or equivalently, the partial differentialequations (3.106), (3.108), (3.110) and (3.96).⇓• There exist functions Θ 0 , Θ j satisfying the second auxiliary system(3.99), (3.100), (3.101) and (3.102).⇓• These solution functions Θ 0 , Θ j satisfy the six families of partial differentialequations (3.69), (3.86), (3.89), (3.91), (3.93) and (3.96).⇓• There exist functions Π j l 1 ,l 2of (x, y), 0 j, l 1 , l 2 m, satisfying thefirst auxiliary system (3.38) of partial differential equations.⇓• There exist functions X, Y j of (x, y) transforming the system y j xx =F j (x, y, y x ), j = 1, . . ., m, to the free particle system Y j XX = 0, j =1, . . ., m.The above last three implications have been already implicitely establishedin the preceding paragraphs, as may be checked by inspectingLemma 3.40 and the formal computations after §3.62.Thus, it remains only to establish the first implication in the above reverselist. Since the second auxiliary system (3.99), (3.100), (3.101) and (3.102)is complete and of first order, a necessary and sufficient condition for theexistence of solutions follows by writing out the following four families ofcross-differentiations:⎧ ( ))y − l 1(3.112)0 = ( Θ 0 x( ) (⎪⎨ 0 = Θ 0 y l 1−y l 20 = ( ) (Θ l 1− x y l 2( )⎪⎩ 0 = Θ l 1−y l 2y l 3Θ 0 y l 1Θ 0 y l 2Θ l 1y l 2,)x(Θ l 1y l 3),x)y l 1y l 2,.

In the hardest techical part of this paper (Section 4 below), we verify thatthese four families of compatibility conditions are a consequence of (I), (II),(III) and (IV). For reasons of space, we shall in fact only study the firstfamily of compatibility conditions, i.e. the first line of (3.112). In the ourmanuscript, we have treated the remaining three families of compatibilityconditions similarly and completely, up to the very end of every branch ofthe coral tree of computations. However, we would like to mention thattypesetting the remaining three cases would add at least fifty pages of Latexto Section 4. Thus, we prefer to expose thoroughly the treatment of the firstfamily of compatibility conditions, explaining implicitely how to guess thetreatment of the remaining three.§4. COMPATIBILITY CONDITIONS FOR THE SECOND AUXILIARYSYSTEMSo, we have to develope the first line of (3.112): we replace Θ 0 x by itsexpression (3.99), we differentiate it with respect to y l 1, we replace Θ 0 by y l 1its expression (3.100), we differentiate it with respect to x and we substract.We get:(4.1)0 = ( ( )Θ 0 x)y − Θ 0 l 1 y l 1x= −2G l 1y l 1y + l 1 Hl 1l 1 ,xy + l 159+ 2 ∑ k− 1 ∑2k+ Θ 0 Θ 0 y l 1 −G k y l 1 Ll 1l1 ,k + 2 ∑ kH k l 1 ,y l 1 Hl 1k− 1 2∑kG k L l 1l 1 ,k,y l 1 − ∑ kH k l 1H l 1k,y l 1 − ∑ kG k y l 1 Lk k,k − ∑ kG k y l 1 Θk − ∑ kG k L k k,k,y l 1 −G k Θ k y l 1 +− 2 3 Ll 1l1 ,l 1 ,xx + 1 3 Hl 1l 1 ,y l 1x −− 2 3 Gl 1 x M l1 ,l 1− 2 3 Gl 1M l1 ,l 1 ,x − 4 3kk∑G k x M l 1 ,k − 4 ∑G k M l1 ,k,x+3kk∑Hl k 1 ,x Ll 1l1 ,k − 1 ∑3+ 1 ∑H l 13 k,xL k l 1 ,l 1+ 1 ∑H l 13 kL k l 1 ,l 1 ,x − 1 3kkkk+ 1 ∑Hl k 2 1 ,x Lk k,k + 1 ∑Hl k 2 1L k k,k,x + 1 ∑Hl k 2 1 ,x Θk + 1 ∑Hl k 2 1Θ k x−− 1 2 L l 1 ,l 1 ,x Θ 0 − 1 2 L l 1 ,l 1Θ 0 x − 1 2 Θ0 x Θl 1− 1 2 Θ0 Θ l 1 x .Here, we underline twice the second order terms. Also, we have underlinedonce the six terms: Θ k y l 1 , Θ0 y l 1 , Θk x , Θ0 x , Θ0 x and Θl 1 x . They must bekkH k l 1L l 1l1 ,k,x +

60replaced by their values given in (3.99), (3.100), (3.101) and (3.102). In thisreplacement, some double sums appear. As before, we use the first indexk = 1, . . .,m for single summation and then the second index p = 1, . . .,mfor double summation. Finally, we put all the second order terms in the firstline, not disturbing the order of appearance of the 73 remaining terms. Weget:(4.2)0 = −2G l 1y l 1y l 1 + 4 3 Hl 1l 1 ,xy l 1 − 2 3 Ll 1l1 ,l 1 ,xx ++ 2 ∑ k− 1 2∑kG k y l 1 Ll 1l1 ,k4H k l 1 ,y l 1 Hl 1k10+ 2 ∑ k− 1 2G k L l 1l 1 ,k,y l 1∑k16H k l 1H l 1k,y l 111− ∑ k− ∑ kG k y l 1 Lk k,k5G k y l 1 Θk α+− ∑ kG k L k k,k,y l 1a−+ ∑ k− 1 2− 1 2G k L k k,k,y l 1∑k∑ka− 2 ∑ kG k L k k,k Ll 1l1 ,l 1+ ∑bkG k L l 1l1 ,l 1Θ k + ∑dkG k M k,l1 ,x18− ∑ k∑G k L p k,l 1L p p,pp∑G k H p k M l 1 ,p23∑G k L p k,l 1Θ p − ∑pkεp− 1 219−∑G k L k k,k Θl 1kG k M k,l1 Θ 0 e− 1 2c−∑G k Θ k Θ l 1kf++ 2 3 Ll 1l1 ,l 1 ,x Θ0 − 1g 3 Hl 1l 1 ,y l 1 Θ0 +h+ 2 3 Gl 1M l1 ,l 1Θ 0 + 4 ∑G k M l1 ,k Θ 0i 3ke− 1 ∑Hl k 2 1L k k,k Θ0 − 1 ∑2klk− 1 3∑kH l 1kL l 1l1 ,l 1Θ 0 j+ 1 3∑Hl k 1Θ k Θ 0 + 1 2 Ll 1l1 ,l 1Θ 0 Θ 0 + 1n 2 Θ0 Θ 0 Θ l 1−omkH k l 1L l 1l1 ,k Θ0 k−− 2 3 Gl 1 x M l1 ,l 11− 2 3 Gl 1M l1 ,l 1 ,x17+ 1 3+ 1 2∑k∑kH l 1k,xL k l 1 ,l 19H k l 1 ,x Lk k,k8+ 1 3+ 1 2∑k− 4 3H l 1kL k l 1 ,l 1 ,x∑Hl k 1L k k,k,xk∑G k x M l 1 ,kk1514− 1 3+ 1 2∑k2− 4 3∑G k M l1 ,k,xkH k l 1 ,x Ll 1l1 ,k∑Hl k 1 ,x Θk γ−k7− 1 3∑k18+H k l 1L l 1l1 ,k,x13+

− 1 3+ 1 3− 1 4+ 1 4∑Hl k 1H k k,y+ 1 ∑H k l k 6 1L k k,k,x + 2 ∑Hl k 3 1G k M k,k +k12k15k20∑ ∑Hl k 1G p M k,p − 1 ∑ ∑Hl k 31Hp k L p k,k+ 1 ∑ ∑Hl k 31H p k Lk k,pk pk pk p2124∑ ∑Hl k 1H p k Lp p,p − 1 ∑ ∑+ 1 ∑Hl k 44 1L k k,k Θ0 l+k∑kp26H k l 1Θ 0 Θ k m− 1 2 Ll 1l1 ,l 1 ,x Θ0 g+kpH k l 1H p k Θp ηk6125−+ G l 1y l 1 Ll 1l1 ,l 1 3− 1 2 Hl 1l1 ,x Ll 1l1 ,l 1 6−− ∑ k+ 1 2G k L l 1l1 ,l 1L l 1l1 ,k∑k22+ 1 2∑kG k L l 1l1 ,l 1L k k,kG k L l 1l1 ,l 1Θ k − 1 4 Ll 1l1 ,l 1Θ 0 Θ 0 n+db+ 1 4∑kH k l 1H l 1kL l 1l1 ,l 127++ G l 1y l 1 Θl −1β1 2 Hl 1l1 ,x Θl 1− ∑ kG k L l 1l1 ,k Θl 1− 1 4 Θ0 Θ 0 Θ l 1+oζ+ 1 2−δ∑G k L k k,k Θl 1kc+ 1 4∑kH k l 1H l 1kΘ l 1θ+ 1 2∑G k Θ k Θ l 1−kf+ 1 3 Hl 1l 1 ,y l 1 Θ0 − 1h 6 Ll 1l1 ,l 1 ,x Θ0 g−− 2 3 Gl 1M l1 ,l 1Θ 0 − 1 ∑G k M l1 ,k Θ 0i 3ke+ 1 ∑Hl k 4 1L k k,k Θ0 + 1 ∑4klk+ 1 3∑kH l 1kL k l 1 ,l 1Θ 0 j− 1 3∑Hl k 1Θ 0 Θ k − 1 4 Ll 1l1 ,l 1Θ 0 Θ 0 − 1n 4 Θ0 Θ 0 Θ l 1.omAs usual, all the terms underlined with the 15 roman alphabetic lettersa, b, . . ., n, o appended vanish evidently. Furthermore, we claim that theeight terms underlined with the 8 Greek alphabetic letters α, β, γ, δ, ε, ζ, ηkH k l 1L l 1l1 ,k Θ0 k+

62and θ also vanish:(4.3)0 =?= − ∑ k+ ∑ kG k y l 1 Θk + G l 1y l 1 Θl 1+ 1 2∑G k L p k,l 1Θ p − ∑pk∑kH k l 1 ,x Θk − 1 2 Hl 1l1 ,x Θl 1+G k L l 1l1 ,k Θl 1− 1 4∑ ∑Hl k 1H p k Θp + 1 4kp∑kH k l 1H l 1kΘ l 1.Indeed, it suffices to observe that this identity coincides with(4.4) 0 = 1 2∑kΘ k ( (3.106)| j:=k; l1 :=l 1 ; l 2 :=l 1).Simplifying then (4.2), we get the explicit formulation of the first family ofcompatibility conditions for the second auxiliary system:(4.5)0 =?= 2G l 1y l 1y l 1 + 4 3 Hl 1l 1 ,xy l 1 − 2 3 Ll 1l1 ,l 1 ,xx −− 2 3 Gl 1 x M l1 ,l 1− 4 3∑+ 2 ∑ G k y l 1 Ll 1l1 ,k − ∑kk− 1 2 Hl 1l1 ,x Ll 1l1 ,l 1− 1 ∑3+ 1 3− 1 3∑kkkH l 1k,xL k l 1 ,l 1− 1 2∑H k k,yH k k l 1−kG k x M l1 ,k + G l 1y l 1 Ll 1l1 ,l 1+G k y l 1 Lk k,k −H k l 1 ,x Ll 1l1 ,k + 1 2∑k∑Hl k 1 ,x Lk k,k +kH k l 1 ,y l 1 Hl 1k− 1 2∑kH l 1k,y l 1 Hk l 1−

− 1 3∑kL l 1l1 ,k,x Hk l 1+ 1 3∑kL k l 1 ,l 1 ,x Hl 1k+ 2 3∑L k k,k,x Hk l 1++ 2 ∑ L l 1l 1 ,k,y l 1 Gk −k− 2 3 M l 1 ,l 1 ,x G l 1− 10 ∑M k,l1 ,x G k −3k− ∑ ∑G k H p k M l 1 ,p + 2 ∑G k Hl k 31M k,k + 1 ∑ ∑G p Hl k 31M k,p −k pkk p− ∑ G k L l 1l1 ,k Ll 1l1 ,l 1+ ∑ ∑G k L p k,l 1L p p,p−kk p− 1 ∑ ∑Hl k 31Hp k L p k,k + 1 ∑ ∑Hl k 31H p k Lk k,p −kp− 1 ∑ ∑Hl k 41H p k Lp p,p + 1 ∑4kpkkpkH k l 1H l 1kL l 1l1 ,l 1.We can now state the main technical lemma of this section and of this paper.Lemma 4.6. The second order partial differential relations (4.5) hold truefor l = 1, . . .,m, and they are a consequence, by differentiations and bylinear combinations, of the fundamental first order partial differential equations(3.106), (3.108), (3.110) and (3.96).4.7. Reconstitution of the appropriate linear combinations. The remainingof Section 4 is entirely devoted to the proof of this statement. Fromthe manual computational point of view, the difficulty of the task is dueto the fact that one has to manipulate formal expressions having from 10to 50 terms. So the real question is: how can we reconstitute the linearcombinations and the differentiations which lead to the goal (4.5) from thedata (3.106), (3.108), (3.110) and (3.96)?.The main trick is to first neglect the first order and the zero order terms inthe goal (4.5). Using the symbol “≡” to denote “modulo first order and thezero order terms”, we formulate the following sub-goal:(4.8) 0 ≡?≡ −2 G l 1y l 1y l 1 + 4 3 Hl 1l 1 ,xy l 1 − 2 3 Ll 1l1 ,l 1 ,xx ,for l 1 = 1, . . .,m. Before estabilishing that these partial differential relationsare a consequence of the data (3.106), (3.108), (3.110) and (3.96)(written with a similar sign ≡), let us check that they are a consequenceof the existence of the change of coordinates (x, y) ↦→ (X, Y ) (however,recall that, in establishing the reverse implications of §3.111, we still donot know that such a change of coordinates really exists); this will confirmthe coherence and the validity of our computations. Importantly, we have63

64been able to achieve systematic corrections of our computations by alwayschecking them alongside with the existence of the change of coordinates(x, y) ↦→ (X, Y ).Coming back to the definition (3.35) and to the approximation (3.58), wehave:(4.9)G l 1= −□ l 1xx∼ = −Yl 1xx ,H l 1l1= −2□ l 1+ xy l 1 □0 ∼ xx = −2 Y l 1+ X xy l 1 xx,L l 1l1 ,l 1= −□ l 1y l 1y l 1 + 2 □0 xy l 1∼ = −Yl 1y l 1y l 1 + 2 X xy l 1.Differentiating the first two lines with respect to y l 1and the third line withrespect to x, and replacing the sign ∼ = by the sign ≡ (in a non-rigorous way,this corresponds essentially to neglecting the derivatives of order 0, 1, 2 and3 of X, Y j and to neglecting the difference between the Jacobian matrix ofthe transformation and the identity matrix), we get:G l 1y l 1y l 1 ≡ −Y l 1xxy l 1y l 1 ,(4.10)H l 1l 1 ,xy l 1 ≡ −2Y l 1xy l 1xy l 1 + X xxxy l 1,L l 1l1 ,l 1 ,xx ≡ −Y l 1y l 1y l 1xx + 2X xy l 1xx .Hence the linear combination −2 · (4.10) 1 + 4 3 · (4.10) 2 − 2 3 (4.10) 3 yieldsthe desired result:(4.11)0 ≡?≡ −2G l 1y l 1y l 1 + 4 3 Hl 1l 1 ,xy l 1 − 2 3 Ll 1l1 ,l 1 ,xx≡ 2Y l 1− 8 xxy l 1y l1 a 3 Y l 1+ 4 xy l 1xy l 1a 3 X xxxy l 1 + 2b 3 Y l 1y l 1y l1xx − 4a 3 X xy l 1xxb≡ 0, indeed!Thanks to this straightforward computation, we guess that the approximatepartial differential relations (4.8) are a consequence of the approximate relations(3.106), (3.108), (3.110) and (3.96), namely:(4.12)(3.106) mod : 0 ≡ −2 G j + 2 y l 1 δj l 1G l 2+ y l 2 Hj l 1 ,x − δj l 1H l 2l2 ,x ,(3.108) mod : 0 ≡ − 1 2 Hj l 1 ,y l 2 + 1 6 δj l 1H l 2l 2 ,y l 2 + 1 3 δj l 2H l 1l 1 ,y l 1 ++ L j l 1 ,l 2 ,x − 1 3 δj l 1L l 2l2 ,l 2 ,x − 2 3 δj l 2L l 1l1 ,l 1 ,x ,(3.110) mod : 0 ≡ L j l 1 ,l 2 ,y l 3 − Lj l 1 ,l 3 ,y l 2 + δj l 3M l1 ,l 2 ,x − δ j l 2M l1 ,l 3 ,x,(3.96) mod : 0 ≡ M l1 ,l 2 ,y l 3 − M l1 ,l 3 ,y l 2.

Here, the sign ≡ means “modulo zero order terms”. Before proceding further,recall the correspondence between partial differential relations:(4.13)(I) = (3.106),(II) = (3.108),(III) = (3.110),(IV) = (3.96).However, these couples of equivalent identities are written slightly differently,as may be read by comparison. To fix ideas and to facilitate the eyecheckingof our subsequent computations, we shall only use and refer to theexact writing of (3.106), of (3.108), of (3.110) and of (3.96).4.14. Construction of a guide. So we want to show that the approximate relation(4.8) is a consequence, by differentiations and by linear combinations,of the approximate identities (4.12). The interest of working with approximateidentities is that formal computations are lightened substantially. Afterhaving discovered which linear combinations and which differentiations areappropriate, i.e. after having constructed a “guide”, in §4.22 below, we shallwrite down the complete computations, including all zero order terms, followingour guide.We shall use two indices l 1 and l 2 with 1 l 1 , l 2 m and, crucially,l 2 ≠ l 1 . Again, the assumption m 2 is used strongly.Firstly, put j := l 1 in (3.106) mod with l 2 ≠ l 1 and differentiate withrespect to y l 1:(4.15) 0 ≡ −2G l 1y l 1y l 1 + 2Gl 2y l 2y l 1 + Hl 1l 1 ,xy l 1 − Hl 2l 2 ,xy l 1 .Secondly, put j := l 2 in (3.106) mod with l 1 ≠ l 2 and differentiate withrespect to y l 2:(4.16) 0 ≡ −2G l 2y l 1y l 2 + Hl 2l 1 ,xy l 2 .Thirdly, put j := l 2 in (3.108) mod with l 1 ≠ l 2 and differentiate with respectto x:(4.17) 0 ≡ − 1 2 Hl 2l 1 ,y l 2x + 1 3 Hl 1l 1 ,y l 1x + Ll 2l1 ,l 2 ,xx − 2 3 Ll 1l1 ,l 1 ,xx .Fourthly, put j := l 1 in (3.108) mod with l 2 ≠ l 1 and differentiate with respectto x:(4.18) 0 ≡ − 1 2 Hl 1l 1 ,y l 2x + 1 6 Hl 2l 2 ,y l 2x + Ll 1l1 ,l 2 ,xx − 1 3 Ll 2l2 ,l 2 ,xx .Fithly, permute the indices (l 1 , l 2 ) ↦→ (l 2 , l 1 ):(4.19) 0 ≡ − 1 2 Hl 2l 2 ,y l 1x + 1 6 Hl 1l 1 ,y l 1x + Ll 2l2 ,l 1 ,xx − 1 3 Ll 1l1 ,l 1 ,xx .65

66Finally, compute the linear combination (4.15)+(4.16)+2·(4.17)−2·(4.19):0 ≡ −2G l 1+ 2G l y l 1y l 2+ H l 1 y l 1y l2 1− H l l 1 a 1 ,xy l 2−1 l 2 2 ,xy l1 b(4.20)− 2G l 2+ H l y l 1y l2 2−la 1 ,xy l2 c− H l 2+ 2 l 1 ,xy l2 c 3 Hl 1+ 2L l 2l 1 ,l 1 ,xy l 1 l1 ,l 2 ,xx − 4 2 d 3 Ll 1l1 ,l 1 ,xx+3+ H l 2− 1 l 2 ,xy l1 b 3 Hl 1− 2L l 2l 1 ,xy l 1 l2 ,l 1 ,xx + 2 2 d 3 Ll 1l1 ,l 1 ,xx.3We indeed get the desired approximate identity:(4.21) 0 ≡ −2 G l 1y l 1y l 1 + 4 3 Hl 1l 1 ,xy l 1 − 2 3 Ll 1l1 ,l 1 ,xx .4.22. Complete computation. Now that the guide is constructed, we canachieve the complete computations.Firstly, put j := l 1 in (3.106) with l 2 ≠ l 1 and differentiate with respectto y l 1:(4.23)0 ≡ −2G l 1y l 1y + l 1 2Gl 2y l 2y + l 1 Hl 1l 1 ,xy − l 1 Hl 2l 2 ,xy + l 1+ 2 ∑ k− 1 ∑2kG k y l 1 Ll 1l1 ,k + 2 ∑ kH k l 1 ,y l 1 Hl 1k− 1 2∑kG k L l 1l 1 ,k,y l 1 − 2 ∑ kH k l 1H l 1k,y l 1 + 1 2∑kG k y l 1 Ll 2l2 ,k − 2 ∑ kH k l 2 ,y l 1 Hl 2k+ 1 2G k L l 2l 2 ,k,y l 1 −∑kH k l 2H l 2k,y l 1 .Secondly, put j := l 2 in (3.106) with l 1 ≠ l 2 and differentiate with respectto y l 2:(4.24)0 ≡ −2G l 2y l 1y + l 2 Hl 2l 1 ,xy + l 22 ∑ kG k y l 2 Ll 2l1 ,k + 2 ∑ kG k L l 2l 1 ,k,y l 2 − 1 2∑kH k l 1 ,y l 2 Hl 2k− 1 2∑kH k l 1H l 2k,y l 2 .

Thirdly, put j := l 2 in (3.108) with l 1 ≠ l 2 and differentiate with respect tox:(4.25)0 ≡ − 1 2 Hl 2l 1 ,y l 2x + 1 3 Hl 1l 1 ,y l 1x + Ll 2l1 ,l 2 ,xx − 2 3 Ll 1l1 ,l 1 ,xx ++ G l 2 x M l1 ,l 2+ G l 2 x M l1 ,l 2 ,x − 2 3 Gl 1 x M l1 ,l 1− 2 3 Gl 1M l1 ,l 1 ,x−− 1 ∑G k x3M l 1 ,k − 1 ∑G k M l1 ,k,x − 1 ∑H l 232 k,xL k l 1 ,l 2− 1 2+ 1 2− 1 3k∑k∑kH k l 1 ,x Ll 2l2 ,k + 1 2H k l 1 ,x Ll 1l1 ,k − 1 3k∑k∑kH k l 1L l 2l2 ,k,x + 1 3H k l 1L l 1l1 ,k,x .k∑kH l 1k,xL k l 1 ,l 1+ 1 3∑k∑k67H l 2kL k l 1 ,l 2 ,x +H l 1kL k l 1 ,l 1 ,x −Fourthly, put j := l 1 in (3.108) with l 2 ≠ l 1 , differentiate with respect to xand permute the indices (l 1 , l 2 ) ↦→ (l 2 , l 1 ):(4.26)0 ≡ − 1 2 Hl 2l 2 ,y l 1x + 1 6 Hl 1l 1 ,y l 1x + Ll 2l2 ,l 1 ,xx − 1 3 Ll 1l1 ,l 1 ,xx ++ G l 2 x M l2 ,l 1+ G l 2M l2 ,l 1 ,x − 1 3 Gl 1 x M l1 ,l 1− 1 3 Gl 1M l1 ,l 1 ,x++ 1 ∑G k x M l1 ,k + 1 ∑G k M l1 ,k,x − 1 ∑H l 2332 k,xL k l 2 ,l 1− 1 2+ 1 2− 1 6k∑k∑kH k l 2 ,x Ll 2l1 ,k + 1 2H k l 1 ,x Ll 1l1 ,k − 1 6k∑k∑kH k l 2L l 2l1 ,k,x + 1 6H k l 1L l 1l1 ,k,x .k∑kH l 1k,xL k l 1 ,l 1+ 1 6∑k∑kH l 2kL k l 2 ,l 1 ,x +H l 1kL k l 1 ,l 1 ,x −

68Finally, compute the linear combination (4.23)+(4.24)+2·(4.25)−2·(4.26):(4.27)0 = −2G l 1y l 1y l 1 + 4 3 Hl 1l 1 ,xy l 1 − 2 3 Ll 1l1 ,l 1 ,xx −− 2 3 Gl 1 x M l1 ,l 1− 4 3+ 2 ∑ k+ ∑ k+ 1 ∑2k− 1 ∑2+ ∑ kG k y l 1 Ll 1l1 ,k − 2 ∑ kHl k 1 ,x Ll 2l2 ,k + 1 ∑3k∑G k x M l 1 ,k+kkH k l 2 ,y l 1 Hl 2k+ 1 2H k l 1H l 1k,y l 1 − 1 2L l 2l2 ,k,x Hk l 1+ 1 3∑G k y l 1 Ll 2l2 ,k + 2 ∑ kH l 1k,xL k l 1 ,l 1− 1 3∑k∑+ 2 ∑ L l 1l 1 ,k,y l 1 Gk + 2 ∑kk− 2 3 M l 1 ,l 1 ,x G l 1− 4 ∑3kkk∑kH l 2k,y l 1 Hk l 2− 1 2H k l 1 ,y l 2 Hl 1k− 1 2L k l 1 ,l 1 ,x Hl 1k− 1 3∑kG k y l 2 Ll 2l1 ,k +H k l 1 ,x Ll 1l1 ,k − ∑ k∑k∑kL l 2l 1 ,k,y l 2 Gk − 2 ∑ kM l1 ,k,x G k .H k l 1 ,y l 1 Hl 1k−H k l 1H l 2k,y l 2 +L l 1l1 ,k,x Hk l 1− ∑ kL l 2l 2 ,k,y l 1 Gk −H k l 2 ,x Ll 2l1 ,k +L l 2l1 ,k,x Hk l 2+In this partial differential relation, importantly, the second order terms areexactly the same as in our goal (4.5). Unfortunately, the first order and thezero order terms are not the same.4.28. Formulation of a new goal. Thus, in order to get rid of the second orderexpression 2 G l 1y l 1y + 4 l 1 3 Hl 1− 2l 1 ,xy l 1 3 Ll 1l1 ,l 1 ,xx, we substract: (4.5)−(4.27).In the result, we write the first order terms in a certain way, adapted in advanceto our subsequent computations. For this substraction yielding (4.29)just below, we have not underlined the terms in (4.5) and in (4.27). However,they may be underlined with a pencil to check that the result (4.29) is

69correct. We get:(4.29)0 =?== − ∑ k+ 2 ∑ kG k y l 1 Lk k,k + Gl 1y l 1 Ll 1l1 ,l 1+ 1 2G k y l 1 Ll 2l2 ,k − ∑ k∑kH k l 1 ,x Ll 2l2 ,k −H k l 1 ,x Lk k,k − 1 2 Hl 1l1 ,x Ll 1l1 ,l 1+− 2 ∑ kG k y l 2 Ll 2l1 ,k + ∑ kH k l 2 ,x Ll 2l1 ,k ++ 1 2− 1 2+ 1 2+ 2 ∑ k∑k∑k∑kH l 2k,y l 2 Hk l 1− 1 3H l 2k,y l 1 Hk l 2+ ∑ kH k l 1 ,y l 2 Hl 2k− 1 2∑H k k,yH k l k 1− ∑kkL l 2l1 ,k,x Hk l 2+∑L l 2l 2 ,k,y l 1 Gk − 2 ∑ kkH k l 2 ,y l 1 Hl 2k+L l 2l 1 ,k,y l 2 Gk − 2 ∑ kL l 2l2 ,k,x Hk l 1+ 2 3M k,l1 ,x G k −∑L k k,k,x Hk l 1−− ∑ ∑G k H p k M l 1 ,p + 2 ∑G k Hl k 31M k,k + 1 ∑ ∑G p Hl k 31M k,p −k pkk p− ∑ G k L l 1l1 ,l 1L l 1l1 ,k + ∑ ∑G k L p k,l 1L p p,p − 1 ∑ ∑Hl k 31Hp k Lp k,k +kk pk p+ 1 ∑ ∑Hl k 31H p k Lk k,p − 1 ∑ ∑Hl k 41H p k Lp p,p + 1 ∑Hl k 4 1H l 1kL l 1l1 ,l 1.kpkWe have underlined plainly the first order terms appearing in lines 1, 2, 3, 4,5, 6 and 7.4.30. Reconstitution of the subgoal (4.29) from (3.106), from (3.108) andfrom (3.110). Now, it suffices to establish that the first order partial differentialrelations (4.29) for 1 l 1 , l 2 m and l 2 ≠ l 1 (crucial assumption)are a consequence of (3.106), of (3.108) and of (3.110) by linear combinations.The auxiliary index l 2 , which is absent in the goal (4.5), will disappearat the end. Differentiations will not be applied anymore. Also, the partialdifferential relations (3.96), which were not used above, will neither be usedin the sequel. However, they are strongly used in the treatment of the remainingthree compatibility conditions (3.112) 2 , (3.112) 3 and (3.112) 4 , thedetail of which we do not copy in the typesetted paper. Finally, the constructionof a guide for the subgoal (4.29) may be guessed similarly as in §4.14above. We shall provide the final computations directly, without any guide:pkk

70they consists of the seven partial differential relations (4.32), (4.33), (4.35),(4.37), (4.39), (4.43) and (4.45) below. At the end, we shall make the addition(4.47) below, producing the desired subgoal (4.29) := (4.32) + (4.33)+ (4.35) + (4.37) + (4.39) + (4.43) + (4.45), with the numerotation of termscorresponding to the order of appearance of the terms of (4.29), as usual.Firstly, put j := k, l 1 := l 1 and l 2 := l 1 in (3.106):(4.31)0 = −2G k y + l 1 2δk l 1G l 1y + l 1 Hk l 1 ,x − δk l 1H l 1l1 ,x ++ 2 ∑ pG p L k l 1 ,p − 2δk l 1∑pG p L l 1l1 ,p − 1 2∑H p l 1Hp k + 1 ∑2 δk l 1ppH p l 1H l 1 p .∑Apply the operator 1 2 k Lk k,k (·) to the preceding equality, namely compute∑12 k Lk k,k · (4.31). This yields:(4.32)0 = − ∑ G k y l 1 Lk k,k + Gl 1y + 1 ∑H l 1 l k 2 1 ,x Lk k,k − 1 2 Hl 1l1 ,x Ll 1l1 ,l 1+kk+ ∑ ∑G p L k k,k Lk l 1 ,p − ∑ G p L l 1l1 ,l 1L k l 1 ,p − 1 ∑ ∑H p4l 1Hp k L k k,k +k ppk p+ 1 ∑H p4l 1H l 1 p L l 1l1 ,l 1.pSecondly, apply the operator − ∑ k Ll 2l2 ,k(·) to (4.32), namely compute− ∑ k Ll 2 · (4.32). This yields:(4.33)0 = 2 ∑ kl2 ,k− 2 ∑ k− 1 ∑2G k y l 1 Ll 2l2 ,k − 2Gl 1y l 1 Ll 2l2 ,l 1− ∑ kp∑pG p L l 2l2 ,k Lk l 1 ,p + 2 ∑ pH p l 1H l 1 p L l 2l2 ,l 1.H k l 1 ,x Ll 2l2 ,k + Hl 1l1 ,x Ll 2l2 ,l 1−G p L l 2l2 ,l 1L l 1l1 ,p + 1 2∑ ∑kpH p l 1H k p L l 2l2 ,k −Thirdly, put j := k, l 1 := l 2 and l 2 := l 1 with l 2 ≠ l 1 in (3.106):(4.34)0 = −2G k y + l 2 2δk l 2G l 1y + l 1 Hk l 2 ,x − δk l 2H l 1l1 ,x ++ 2 ∑ pG p L k l 2 ,p − 2δk l 2∑pG p L l 1l1 ,p − 1 2∑H p l 2Hp k + 1 ∑2 δk l 2ppH p l 1H l 1 p .

Next, apply the operator ∑ k Ll 2l1 ,k (·) to (4.34), namely compute ∑ k Ll 2l1 ,k ·(4.34). This yields:(4.35)0 = −2 ∑ k+ 2 ∑ k+ 1 ∑2pG k y l 2 Ll 2l1 ,k + 2Gl 1y l 1 Ll 2l1 ,l 2+ ∑ k∑pG p L k l 2 ,p Ll 2l1 ,k − 2 ∑ pH p l 1H l 1 p L l 2l1 ,l 2.H k l 2 ,x Ll 2l1 ,k − Hl 1l1 ,x Ll 2l1 ,l 2+G p L l 1l1 ,p Ll 2l1 ,l 2− 1 2Fourthly, put j := l 2 , l 1 := k and l 2 := l 2 in (3.108):(4.36)0 = − 1 2 Hl 2k,y l 2 + 1 6 δl 2kH l 2l 2 ,y l 2 + 1 3 Hk k,y k + L l 2k,l2 ,x −− 1 3 δk l 2L l 2l2 ,l 2 ,x − 2 3 Lk k,k,x ++ G l 2M k,l2 − 1 3 δk l 2G l 2M l2 ,l 2− 2 3 Gk M k,k + 1 3 δl 2k∑− 1 ∑G p M k,p − 1 ∑32p− 1 6 δl 2k∑ppH p l 2L l 2l2 ,p + 1 3H l 2 p L p k,l 2+ 1 2p∑pp∑ ∑kpG p M l2 ,p−H p k Ll 2l2 ,p + 1 6 δl 2k∑∑Hp k L p k,k − 1 ∑H p3k Lk k,p .p71H p l 2H k p L l 2l1 ,k +pH l 2 p L p l 2 ,l 2−Next, apply the operator − ∑ k Hk l 1(·) to (4.36), namely compute − ∑ k Hk l 1·(4.36). This yields:(4.37)0 = 1 2∑k− ∑ k− ∑ k− 1 ∑3− 1 2H l 2k,y l 2 Hk l 1− 1 6 Hl 2l 2 ,y l 2 Hl 2l1− 1 3pL l 2k,l2 ,x Hk l 1+ 1 3 Ll 2l2 ,l 2 ,x Hl 2l1+ 2 3G l 2H k l 1M k,l2 + 1 3 Gl 2H l 2l1M l2 ,l 2+ 2 3∑ ∑kG p H l 2l1M l2 ,p + 1 3pkH k l 1H p k Ll 2l2 ,p − 1 6p∑H k k,yH k k l 1−k∑L k k,k,x Hk l 1−k∑G k Hl k 1M k,k −∑ ∑G p Hl k 1M k,p + 1 ∑ ∑2∑pkH l 2l1H l 2 p L p l 2 ,l 2+ 1 6− 1 ∑ ∑Hl k 31Hp k L p k,k + 1 ∑ ∑Hl k 31H p k Lk k,p .kpkpk∑ppH k l 1H l 2 p L p k,l 2−H l 2l1H p l 2L l 2l2 ,p −

72Fithly, put j := l 2 , l 1 := k and l 2 := l 1 in (3.108):(4.38)0 = − 1 2 Hl 2k,y l 1 + 1 6 δl 2kH l 1l 1 ,y l 1 + Ll 2k,l1 ,x − 1 3 δl 2kL l 1l1 ,l 1 ,x ++ G l 2M k,l1 − 1 3 δl 2kG l 1M l1 ,l 1+ 1 3 δl 2k∑+ 1 2∑pH p k Ll 2l1 ,p + 1 6 δl 2k∑ppG p M l1 ,p − 1 2H l 1 p L p l 1 ,l 1− 1 6 δl 2k∑p∑pH p l 1L l 1l1 ,p .H l 2 p L p k,l 1+Next, apply the operator ∑ k Hk l 2(·) to (4.38), namely compute ∑ k Hk l 2 ·(4.38). This yields:(4.39)0 = − 1 2∑kH l 2k,y l 1 Hk l 2+ 1 6 Hl 1l 1 ,y l 1 Hl 2l2+ ∑ k− 1 3 Ll 1l1 ,l 1 ,x Hl 2l2++ ∑ k− 1 ∑ ∑2− 1 6kG l 2H k l 2M k,l1 − 1 3 Gl 1H l 2l2M l1 ,l 1+ 1 3∑ppH k l 2H l 2 p L p k,l 1+ 1 2H l 2l2H p l 1L l 1l1 ,p .∑ ∑kpL l 2k,l1 ,x Hk l 2−∑pG p H l 2l2M l1 ,p−H k l 2H p k Ll 2l1 ,p + 1 6∑pH l 2l2H l 1 p L p l 1 ,l 1−Sixthly, we form the expression:(4.40) (3.108)| j:=k; l1 :=l 1 ; l 2 :=l 2− (3.108)| j:=k; l1 :=l 2 ; l 2 :=l 1.

73Writing term by term the substractions, we get:(4.41)0 = − 1 2 Hk l 1 ,y l 21+ 1 2 Hk l 2 ,y l 12+ 1 6 δk l 1H l 2l 2 ,y l 2 3− 1 6 δk l 2H l 1l 1 ,y l 1 4++ 1 3 δk l 2H l 1l 1 ,y l 1 4− 1 3 δk l 1H l 2l 2 ,y l 2 3+ L k l 1 ,l 2 ,x a − Lk l 2 ,l 1 ,x a −− 1 3 δk l 1L l 2l2 ,l 2 ,x5+ 1 3 δk l 2L l 1l1 ,l 1 ,x6− 2 3 δk l 2L l 1l1 ,l 1 ,x6+ 2 3 δk l 1L l 2l2 ,l 2 ,x5++ G k M l1 ,l 2 b − Gk M l2 ,l 1 b − 1 3 δk l 1G l 2M l2 ,l 27+ 1 3 δk l 2G l 1M l1 ,l 18−− 2 3 δk l 2G l 1M l1 ,l 1+ 28 3 δk l 1G l 2M l2 ,l 2+ 1 ∑7 3 δk l 1G p M l2 ,p− 1 ∑3 δk l 2G p M l1 ,p+ 1 2p∑H p l 1L k l 2 ,pp− 1 6 δk l 1∑p− 1 3 δk l 1∑p11H p l 2L l 2l2 ,p10− 1 215H l 2 p L p l 2 ,l 213+ 1 ∑3 δk l 1G p M l2 ,pp∑H p l 2L k l 1 ,pp+ 1 6 δk l 2∑p− 1 3 δk l 2∑p12H p l 1L l 1l1 ,pH p l 1L l 1l1 ,p9p− 1 2+ 1 6 δk l 1∑1616p∑p+ 1 3 δk l 2∑9H k p Lp l 1 ,l 2cH l 2 p L p l 2 ,l 213p+ 1 3 δk l 1∑p− 1 ∑3 δk l 2G p M l1 ,p+ 1 2p10−∑Hp k Lp l 2 ,l 1c+p− 1 6 δk l 2∑H l 1 p L p l 1 ,l 114−H p l 2L l 2l2 ,p15.pH l 1 p L p l 1 ,l 114−Simplifying, we get:(4.42)0 = − 1 2 Hk l 1 ,y l 2 + 1 2 Hk l 2 ,y l 1 − 1 6 δk l 1H l 2l 2 ,y l 2 + 1 6 δk l 2H l 1l 1 ,y l 1 ++ 1 3 δk l 1L l 2l2 ,l 2 ,x − 1 3 δk l 2L l 1l1 ,l 1 ,x ++ 1 3 δk l 1G l 2M l2 ,l 2− 1 3 δk l 2G l 1M l1 ,l 1+ 2 ∑3 δk l 1G p M l2 ,p − 2 ∑3 δk l 2G p M l1 ,p++ 1 ∑H p2l 1L k l 2 ,p − 1 ∑H p2l 2L k l 1 ,p − 1 ∑6 δk l 1p+ 1 6 δk l 1∑pH p l 2L l 2l2 ,p − 1 6 δk l 2∑ppH p l 1L l 1l1 ,p .ppH l 2 p L p l 2 ,l 2+ 1 6 δk l 2∑ppH l 1 p L p l 1 ,l 1+

74Next, apply the operator − ∑ k Hl 2k(·) to (4.42), namely compute− ∑ k Hl 2k · (4.42). This yields:(4.43)0 = 1 ∑H k2l 1 ,y l 2 Hl 2k− 1 ∑H k2l 2 ,y l 1 Hl 2k+ 1 6 Hl 2l 2 ,y l 2 Hl 2l1−kk− 1 6 Hl 1l 1 ,y l 1 Hl 2l2− 1 3 Ll 2l2 ,l 2 ,x Hl 2l1+ 1 3 Ll 1l1 ,l 1 ,x Hl 2l2−− 1 3 Gl 2H l 2l1M l2 ,l 2+ 1 3 Gl 1H l 2l2M l1 ,l 1− 2 3+ 2 3+ 1 6+ 1 6∑p∑p∑pG p H l 2l2M l1 ,p − 1 2H l 2l1H l 2 p L p l 2 ,l 2− 1 6H l 2l2H p l 1L l 1l1 ,p .∑ ∑k∑pp∑pH l 2kH p l 1L k l 2 ,p + 1 2H l 2l2H l 1 p L p l 1 ,l 1− 1 6G p H l 2l1M l2 ,p+∑p∑ ∑Seventhly, put j := l 2 , l 1 := k, l 2 := l2 and l 3 := l 1 in (3.108):(4.44)0 = L l 2k,l 2 ,y − l 1 Ll 2k,l 1 ,y − M l 2 k,l 1 ,x++ 1 2 Hl 2l1M k,l2 − 1 2 Hl 2l2M k,l1 + 1 2 δl 2k∑− 1 2∑H p k M l 1 ,p + ∑pppL p k,l 1L l 2l2 ,p − ∑ pkpH l 2kH p l 2L k l 1 ,p +H l 2l1H p l 2L l 2l2 ,p +H p l 1M l2 ,p − 1 2 δl 2k∑L p k,l 2L l 2l1 ,p ,pH p l 2M l1 ,p−and then apply the operator 2 ∑ k Gk (·):(4.45)0 = 2 ∑ L l 2k,l 2 ,y l 1 Gk − 2 ∑ L l 2k,l 1 ,y l 2 Gk − 2 ∑kkkM k,l1 ,x G k ++ ∑ k− ∑ p− 2 ∑ kG k H l 2l1M l1 ,l 2− ∑ kG l 2H p l 2M l1 ,p − ∑ k∑pG k L p k,l 2L l 2l1 ,p .Finally, achieve the additionG k H l 2l2M k,l1 + ∑ G l 2H p l 1M l2 ,p−p∑G k H p k M l 1 ,p + 2 ∑ ∑G k L p k,l 1L l 2l2 ,p −pk p(4.46) (4.32) + (4.33) + (4.35) + (4.37) + (4.39) + (4.43) + (4.45).We copy these seven formal expression, we underline the vanishing termsand we number the remaining terms so as to respect the order of appearance

of the terms of the subgoal (4.29):(4.47)0 = − ∑ k+ ∑ k+ 1 4G k y l 1 Lk k,k∑G p L k k,k Lk l 1 ,pp∑p1+ G l 1y l 1 2+ 1 224H p l 1H l 1 p L l 1l1 ,l 128+− ∑ p∑kH k l 1 ,x Lk k,k3G p L l 1l1 ,l 1L k l 1 ,p23− 1 2 Hl 1l1 ,x Ll 1l1 ,l 1 4+− 1 475∑ ∑H p l 1Hp k Lk k,kkp27++ 2 ∑ k− 2 ∑ k− 1 ∑2G k y l 1 Ll 2l2 ,kp∑p5− 2G l 1y l 1 Ll 2l2 ,l 1− ∑akG p L l 2l2 ,k Lk l 1 ,pH p l 1H l 1 p L l 2l2 ,l 1j−g+ 2 ∑ pH k l 1 ,x Ll 2l2 ,kG p L l 2l2 ,l 1L l 1l1 ,p6h+ H l 1l1 ,x Ll 2l2 ,l 1 b −+ 1 2∑ ∑kpH p l 1H k p L l 2l2 ,ki−− 2 ∑ k+ 2 ∑ k+ 1 ∑2G k y l 2 Ll 2l1 ,kp∑p7+ 2G l 1y l 1 Ll 2l1 ,l 2+ ∑akG p L k l 2 ,p Ll 2l1 ,kH p l 1H l 1 p L l 2l1 ,l 2j+k− 2 ∑ pH k l 2 ,x Ll 2l1 ,kG p L l 1l1 ,p Ll 2l1 ,l 2h− 1 28− H l 1l1 ,x Ll 2l1 ,l 2b +∑ ∑kpH p l 2H k p L l 2l1 ,kl+

76+ 1 2∑k− ∑ k− ∑ kH l 2k,y l 2 Hk l 19L l 2k,l2 ,x Hk l 111− 1 6 Hl 2l 2 ,y l 2 Hl 2l1c− 1 3+ 1 3 Ll 2l2 ,l 2 ,x Hl 2l1d+ 2 3G l 2H k l 1M k,l2m+ 1 3 Gl 2H l 2l1M l2 ,l 2n+ 2 3∑H k k,yH k k l 1−10k∑L k k,k,x Hk l 1−12k∑G k Hl k 1M k,k −21− 1 ∑G p H l 23 l1M l2 ,p + 1 ∑ ∑G p Hl k 31M k,p + 1 ∑ ∑2pk pk po22− 1 ∑ ∑Hl k 21H p k Ll 2l2 ,p− 1 ∑H l 26 l1H l 2 p L p l 2 ,l 2+ 1 ∑6k pppiq− 1 ∑ ∑Hl k 31Hp k L p k,k+ 1 ∑ ∑Hl k 31H p k Lk k,p −kp25kpk26H k l 1H l 2 p L p k,l 2p−H l 2l1H p l 2L l 2l2 ,pr−− 1 2∑kH l 2k,y l 1 Hk l 213+ 1 6 Hl 1l 1 ,y l 1 Hl 2l2e+ ∑ kL l 2k,l1 ,x Hk l 214−− 1 3 Ll 1l1 ,l 1 ,x Hl 2l2f++ ∑ k− 1 2− 1 6G l 2Hl k 2M k,l1 − 1 3 Gl 1H l 2l2M l1 ,l 1+ 1t 3s∑ ∑k∑ppHl k 2H l 2p L p k,l 1+ 1 2vH l 2l2H p l 1L l 1l1 ,px+∑ ∑kp∑pH k l 2H p k Ll 2l1 ,pG p H l 2l2M l1 ,pl+ 1 6∑pu−H l 2l2H l 1p L p l 1 ,l 1w−

77+ 1 2∑kH k l 1 ,y l 2 Hl 2k15− 1 2∑kH k l 2 ,y l 1 Hl 2k16+ 1 6 Hl 2l 2 ,y l 2 Hl 2l1c−− 1 6 Hl 1l 1 ,y l 1 Hl 2l2− 1e 3 Ll 2l2 ,l 2 ,x Hl 2l1+ 1d 3 Ll 1l1 ,l 1 ,x Hl 2l2f−∑− 1 3 Gl 2H l 2l1M l2 ,l 2+ 1n 3 Gl 1H l 2l2M l1 ,l 1− 2t 3+ 2 3+ 1 6+ 1 6∑p∑p∑pG p H l 2l2M l1 ,pu− 1 2H l 2l1H l 2 p L p l 2 ,l 2− 1 6qH l 2l2H p l 1L l 1l1 ,px+∑ ∑k∑pppH l 2kH p l 1L k l 2 ,pG p H l 2l1M l2 ,ppH l 2l2H l 1 p L p l 1 ,l 1− 1 6w+ 1 2∑po∑ ∑kp+H l 2l1H p l 2L l 2l2 ,pH l 2kH p l 2L k l 1 ,pr+v++ 2 ∑ kL l 2k,l 2 ,y l 1 Gk 17− 2 ∑ kL l 2k,l 1 ,y l 2 Gk 18− 2 ∑ kM k,l1 ,x G k 19++ ∑ k− ∑ p− 2 ∑ kG k H l 2l1M l1 ,l 2− ∑okG l 2H p l 2M l1 ,p∑ps− ∑ kG k L p k,l 2L l 2l1 ,pkG k H l 2l2M k,l1 + ∑ G l 2H p l 1M l2 ,p −pum∑G k H p k M l 1 ,p + 2 ∑ ∑G k L p k,l 1L l 2l2 ,ppk p.In conclusion, there is exact coincidence with the subgoal (4.29). Theproof that the first family (3.112) 1 of compatibility conditions of the secondauxiliary system (3.99), (3.100), (3.101) and (3.102) are a consequence of(I), (II), (III) and (IV) of Theorem 1.7 (3) is complete. Granted that thetreatment of the other three families of compatibility conditions (3.112) 2 ,(3.112) 3 and (3.112) 4 is similar (and as well painful), we consider that theproof of the equivalence between (1) and (3) in Theorem 1.7 is complete,now.20g−§5. GENERAL FORM OF THE POINT TRANSFORMATIONOF THE FREE PARTICLE SYSTEMThis section is devoted to the exposition of a complete proof ofLemma 3.32. To start with, we must develope the fundamental equations(3.10), for j = 1, . . .,m. Recalling that the total differentiation

78operator is given by D = ∂∂x + ∑ ml 1 =1 yl 1compute first(5.1) ⎧DDX = D ⎪⎨⎪⎩[X x += X xx + 2m∑l 1 =1m∑l 1 =1y l 1x · X y l 1]y l 1x · X xy l 1 +m∑x ·l 1 =1 l 2 =1∂+ ∑ m∂y l 1 l 1 =1 yl 1 xx ·m∑y l 1x y l 2x · X y l 1y l 2 +∂∂y l 1 x, wem∑l 1 =1y l 1xx · X y l 1 ,and(5.2) ⎧DDY ⎪⎨j = D⎪⎩[Y jx += Y jxx + 2m∑l 1 =1m∑l 1 =1y l 1x · Y jy l 1]y l 1x · Y jxy l 1 + m∑m∑l 1 =1 l 2 =1y l 1x y l 2x · Y jy l 1y l 2 + m∑l 1 =1y l 1xx · Y jy l 1 .Now, we can develope the equation 0 = −DY j · DDX + DX · DDY j ,which yields(5.3)0 = −+[[Y jx ++X x ++m∑l 1 =1m∑l 1 =1 l 2 =1m∑l 1 =1m∑y l 1x · Y jy l 1m∑]·[X xx + 2y l 1x y l 2x · X y l 1y l 2 +y l 1x · X y l 1]·m∑l 1 =1 l 2 =1[m∑l 1 =1m∑l 1 =1Y jxx + 2 m∑l 1 =1y l 1x y l 2x · Y jy l 1y l 2 + m∑l 1 =1y l 1x · X xy l 1 +y l 1xx · X y l 1]+y l 1x · Y jxy l 1 +y l 1xx · Y jy l 1]=

79= − X xx Yx j + Y jm∑+++++l 1 =1m∑y l 1x ·m∑l 1 =1 l 2 =1m∑m∑l 1 =1 l 2 =1 l 3 =1m∑l 1 =1m∑y l 1xx ·m∑l 1 =1 l 2 =1xx X x +[−2 X xy l 1 Yx j + 2 Y j X xy l 1 x−]−X xx Y j + Y jy l 1 xx X y l 1 +[y l 1x y l 2x · −X y l 1y l 2 Yx j + Y jy l 1y X l 2 x−m∑]−2 X xy l 2 Y j + 2 Y j X y l 1 xy l 2 y l 1 +[y l 1x y l 2x y l 3x · −X y l 2y l 3 Y j + Y jy l 1 y l 2y X l 3 y l 1[−X y l 1 Y jx + Y jy l 1 X x]+[]y l 1xx y l 2x · −X y l 1 Y j + Y j X y l 2 y l 1 y l 2 .]+The goal is to show that after solving these m equations for j = 1, . . .,mwith respect to the yxx l , l = 1, . . .,m, one obtains the expression (3.33) ofLemma 3.32, or equivalently, using the ∆ notation instead of the squarenotation, one obtains(5.4)⎧0 = yxx j · ∆ ( x|y 1 | · · · |y m) + ∆ ( x|y 1 | · · · | j xx| · · · |y m) +m∑+ y l 1x · [2∆ ( x|y 1 | · · · | j xy l 1| · · · |y m) −⎪⎨⎪⎩++l 1 =1m∑m∑l 1 =1 l 2 =1m∑m∑l 1 =1 l 2 =1 l 3 =1−δ j l 1∆ ( xx|y 1 | · · · |y m)] +y l 1x y l 2x · [∆ ( x|y 1 | · · · | j y l 1y l 2| · · · |y m) −m∑−2 δ j l 1∆ ( xy l 2|y 1 | · · · |y m)] +y l 1x y l 2x y l 3x[−δjl 1∆ ( y l 2y l 3|y 1 | · · · |y m)] .Unfortunately, the equations (5.3) are not solved with respect to the yxx,jbecause in its last line, we notice that the y l 2 xx are mixed with the y l 1 x . Consequently,we have to solve a linear system of m equations with the unknowns

80yxx j of the form(5.5) {m∑0 = A j +l 1 =1y l 1xx ·[−X y l 1 Y jx + Y jy l 1 X x +m∑l 2 =1[]]y l 2x · −X y l 1 Y j y l 2 y l 1 y l 2 , ,for j = 1, . . .,m, where A j is an abbreviation for the terms appearing inthe lines 5, 6, 7, 8, 9 and 10 of (5.3), or even more compactly, changing theindex j to the index k{(5.6)0 = A k +m∑l 1 =1y l 1xx · B k l 1,for k = 1, . . .,m, where Bl k 1is an abbreviation for the terms in the bracketsin (5.5).Thanks to the assumption that the determinant (3.2) is the identity determinantat (x, y) = (0, 0), we deduce that the determinant of the m × mmatrix (Bl k 1) 1km1l 1 m is also the identity determinant at (x, y, y x) = (0, 0, 0). Itfollows that the determinant of the m × m matrix (Bl k 1) 1km1l 1 mis nonvanishingin a neighborhood of the origin in the first order jet space. Consequently,we can apply the rule of Cramer to solve the yxx j explicitely interms of theA k and of the Bl k 1as follows⎧ ∣ ∣∣∣∣∣ B1 1 · · · A 1 · · · B 1 m· · · · · · · · · · · · · · ·⎪⎨(5.7) yxx j = − B1 m · · · A m · · · Bmm ∣B1 1 · · · B 1 · · · B 1 m⎪⎩· · · · · · · · · · · · · · ·∣ B1 m · · · B m · · · Bmm ∣where on the numerator, the only modification of the determinant of the matrix(Bl k 1) 1km1l 1 mis the replacement of its j-th column by the column vectorA. We have to show that after replacing the A k and the Bl k 1by their completeexpressions, one indeed obtains the desired equation (5.4). As in (3.43), weshall introduce a notation for the two m×m determinants appearing in (5.6):we write this quotient under the form{ ∣ ∣B k(5.8) yxx j = − 1 | · · · | j A k | · · · |Bmk ∣ ∣ ∣ Bk1 | · · · | j Bj k| · · · |Bk ∣ ∣ ,mwhere it is understood that B k 1, . . .,B k j , . . ., B k m and A k are column vectorswhose index k (for their lines) varies from 1 to m. This representation ofdeterminants emphasizing only its columns will be appropriate for later manipulations.

Our first task is to compute the determinant in the denominator of (5.8).Recalling that we make the notational identification y 0 ≡ x, it will be convenientto reexpress the B k l 1in a slightly compacter form, using the totaldifferentiation operator D:(5.9) ⎧⎪⎨⎪⎩B k l 1= −X y l 1 Y kx + Y ky l 1X x +m∑l 2 =1= Y ky l 1· DX − X y l 1 · DY k .[]y l 2x · −X y l 1 Y ky l 2+ Y ky l 1X y l 2 =81Lemma 5.10. We have the following expression for the determinant of thematrix (B k l 1) 1km1l 1 m :(5.11){ ∣ ∣ ∣ ∣Y ky 1 · DX − X y 1 · DY k | · · · |Y ky m · DX − X y m · DY k∣ ∣ ∣ ∣ == [DX] m−1 · ∆ ( x|y 1 | · · · |y m) .Proof. By multilinearity, we may develope the determinant written in thefirst line of (5.11). Since it contains two terms in each columns, we shouldobtain a sum of 2 m determinants. However, since the obtained determinantsvanish as soon as the column DY k (multiplied by various factors X y l) appearsat least two different places, it remains only (m + 1) nonvanishingdeterminants, those for which the column DY k appears at most once:(5.12)⎧∣ ∣ Yk⎪⎨ y · DX − X 1 y 1 · DY k | · · · |Yy k · DX − X m y m · DY k∣ ∣ == [DX] m · ∣∣ ∣ ∣ ∣Y k ky1| · · · |Yy ∣ − [DX] m−1 X⎪⎩m y 1 · ∣∣ ∣ ∣ ∣DY k |Y k ky2| · · · |Yy − · · · −m− [DX] m−1 X y m · ∣∣ ∣ Yk ky1| · · · |Yy m−1|DY k∣ ∣ .To establish the desired expression appearing in the second line of (5.11),we factor out by [DX] m−1 and we develope all the remaining total differentiationoperators D. Since y 0 ≡ x, we have yx 0 = 1, and this enables us tocontract X x + ∑ ml 1y l 1 x X y l 1 as ∑ ml 1 =0 yl 1 x X y l 1 . So, we achieve the following

82computation (further explanations and comments just afterwards):(5.13) ⎧ {∑ m= [DX] m−1 · y l 1x X y l 1 · ∣∣ ∣ ∣ ∣Y k k1| · · · |Yy ∣ − m⎪⎨⎪⎩= [DX] m−1 ·l 1 =0−{ m∑m∑l 1 =0− · · · −l 1 =0y l 1y l 1x X y 1 ·m∑l 1 =0y∣ ∣∣∣Y ky l 1|Y k k ∣∣ ∣∣y2| · · · |Yy − my l 1x X y m ·∣∣∣Y k k1| · · · |Yyy m−1|Y ky l 1x X y l 1 · ∣∣ ∣ ∣ ∣Y k k1| · · · |Yy ∣ − X m y 1 · ∣∣ ∣ ∣Y ky}∣x |Y ky∣ k2| · · · |Yy ∣ − m−yx 1 X y 1 · ∣∣ ∣ Yky 1|Y ∣ k ky2| · · · |Yy ∣ − · · · −m−X y m · ∣∣ ∣ Yk ky1| · · · |Yy m−1|Y xk ∣ ∣ − ymx X y m · ∣∣ ∣ Yk ky1| · · · |Y= [DX] m−1 · {Xx · ∣∣ ∣ ∣ Yk ky1| · · · |Y ∣y − m Xy 1 · ∣∣ ∣ ∣ Ykx |Y k ky2| · · · |Y ∣y − · · · −m−X y m · ∣∣ ∣ Yky 1| · · · |Yy k m−1|Y xk ∣ ∣ }= [DX] m−1 · {∆(x|y 1 | · · · |y m ) } .For the passage to the equality of line 4, using the fact that a determinanthaving two identical columns vanishes, we observe that in each of the msums ∑ ml 1 =0appearing in lines 2 and 3 (including the cdots), there remainsonly two non-vanishing determinants. For the passage to the equality of line7, we just sum up all the linear combinations of determinants appearing inlines 4, 5 and 6. Finally, for the passage to the equality of line 9, we recognizethe development of the fundamental Jacobian determinant (3.2) alongits first line (X x , X y 1, . . .,X y m), modulo some permutations of columns inthe m × m minors. The proof is complete.Our second task, similar but computationnally more heavy, is to computethe determinant in the numerator of (5.8). First of all, we have to re-expressthe A k defined implicitely between (5.3) and (5.5) using the total differentiationoperator to contract them as follows(5.14)⎧⎪⎨⎪⎩A k = DX · Y kxx − DY k · X xx + 2+m∑m∑l 1 =1 l 2 =1m∑l 1 =1y m−1|Y k[]y l 1x · DX · Y kxy l 1− DY k · X xy l 1 +[]y l 1x y l 2x · DX · Y ky l 1y l 2− DY k · X y l 1y l 2 .y m ∣ ∣ ∣ ∣}

Replacing this expression of A k in (5.8), taking account of the expression ofthe denominator already obtained in the second line of (5.11) and abbreviating∆(x|y 1 | · · · |y m ) as ∆, we may write (5.8) in length and then develope itby linearity as follows(5.15)⎧⎪⎨⎪⎩y j xx ==−1[DX] m−1 · ∆ ·−1[DX] m−1 · ∆ ·⎡⎢⎣⎡⎢⎣∣ ∣ ∣ Yky 1 · DX − X y 1 · DY k | · · ·· · · | j DX · Yxx k − DY k · Y km∑+2+l 1 =1m∑y l 1x ·m∑l 1 =1 l 2 =1∣ ∣ ∣ Yky 1 · DX − X y 1 · DY k | · · ·+ 2+xx+83[DX · Y kxy l 1− DY k · X xy l 1]+⎥· · · |Yy k · DX − X m y m · DY k∣ ∣ ∣ ⎦⎤[]y l 1x y l 2x · DX · Y ky l 1y − DY k · X l 2 y l 1y l 2 | · · ·· · · | j DX · Y kxx − DY k · X xx | · · ·m∑l 1 =1· · · |Y ky m · DX − X y m · DY k∣ ∣ ∣ ∣ +y l 1x · ∣∣ ∣ ∣ Yky 1 · DX − X y 1 · DY k | · · ·· · · | j DX · Y kxy l 1 − DY k · X xy l 1 | · · ·m∑l 1 =1 l 2 =1· · · |Yy k · DX − X m y m · DY k∣ ∣ ∣ +m∑y l 1x y l 2x · ∣∣ ∣ Yky · DX − X 1 y 1 · DY k | · · ·· · · | j DX · Y ky l 1y l 2− DY k · X y l 1y l 2 | · · ·· · · |Y ky m · DX − X y m · DY k∣ ∣ ∣ ∣As it is delicate to read, let us say that lines 2, 3 and 4 just express the j-thcolum | j A k | of the determinant |B 1 k| · · · |j A k | · · · |Bm k |, after replacementof A k by its complete expression (5.14).In lines 6, 7, 8; in lines 9, 10, 11; and in lines 12, 13, 14, there arethree families of m×m determinants containing a linear combination (soustraction)having exactly two terms in each column. As in the proof ofLemma 5.10, by multilinarity, we have to develope each such determinant.In principle, for each development, we should get 2 m terms, but since theobtained determinants vanish as soon as the column DY k (modulo a multiplicationby some factor) appears at least twice, it remains only (m+1) nonvanishingdeterminants, those for which the column DY k appears at most.⎥⎦⎤

84once. In addition, for each of the obtained determinant, the factor [DX] m−1appears (sometimes even the factor [DX] m ), so that this factor compensatesthe factor [DX] m−1 in the numerator. In sum, the continuation of the hugecomputation yields:(5.16)⎡DX · ∣∣ ∣ Yky 1| · · · |j Yxx k | · · · |Y ∣ y k∣ − myxx j = − ∆·1 ⎢⎣− X y 1 · ∣∣ ∣ ∣ DY k | · · · | j Yxx| k · · · |Yy k∣ − · · · −m− X xx · ∣∣ ∣ Yky 1| · · · ∣ |j DY k | · · · |Yy k∣ − · · · −m− X y m · ∣∣ ∣ Yky 1| · · · |j Yxx k | · · · |DY k∣ ∣ ∣ +m∑+ 2 y l 1∣ ∣x DX ·∣Y ky 1| · · · |j Y kxy | · · · |Y k ∣∣ ∣∣ l 1 y −m l 1 =1m∑∣ − 2 y l 1∣x X y 1 ·∣DY k | · · · | j Y kxy l 1| · · · |Yy k ∣∣ ∣∣ − · · · −m l 1 =1m∑− 2 y l 1x X xy l 1 · ∣∣ ∣ Yky 1| · · · ∣ |j DY k | · · · |Yy k∣ − · · · −m l 1 =1m∑∣ .− 2 y l 1∣x X y m ·∣Y ky 1| · · · |j Y kxy l 1| · · · |DY k ∣∣ ∣∣+l 1 =1m∑ m∑+ y l 1x y l 2∣ ∣x DX · ∣∣Y ky 1| · · · |j Y ky l 1y l 2| · · · |Yy k ∣∣ ∣∣ −m l 1 =1 l 2 =1m∑ m∑− y l 1x y l 2∣ ∣x X y 1 · ∣∣DY k | · · · | j Y ky l 1y | · · · |Y k ∣∣ ∣∣ l 2 y − · · · −m l 1 =1 l 2 =1m∑ m∑− y l 1x y l 2x X y l 1y l 2 · ∣∣ ∣ Yky 1| · · · ∣ |j DY k | · · · |Yy k− · · · −m l 1 =1 l 2 =1m∑ m∑− y l 1x y l 2∣ ⎥∣x X y m · ∣∣Y ky 1| · · · |j Y ky l 1y l 2| · · · |DY k ∣∣ ∣∣ ⎦l 1 =1 l 2 =1⎤To establish the desired expression (5.4), we must develope all the total differentiationoperators D of the terms DX placed as factor and of the termsDY k placed in various columns of determinants. We notice that in developingDY k , we obtain columns Y ky(multiplied by the factor y l l x ) and for onlythree (or two) values of l = 0, 1, . . ., m, this column does not already appearin the corresponding determinant, so that (m − 1) determinants vanish andonly 3 (or 2) remain nonzero. Taking account of these simplifications, we

85have the continuation(5.17) −y j xx · ∆ = I + II + III,where the term I is the development of lines 1, 2, 3, 4 of (5.16); the term II isthe development of lines 5, 6, 7, 8 of (5.16); and the term III is the developmentof lines 9, 10, 11, 12 of (5.16). So we get firstly (further explanationsfollows):m∑l=0y l x X y l · ∣∣ ∣ ∣ Yky 1| · · · |j Y kxx| · · · |Y ky m ∣ ∣ ∣ ∣1−− X y 1 · ∣∣ ∣ ∣ Ykx | · · · | j Y kxx | · · · |Y ky m ∣ ∣ ∣ ∣ −− y 1 x X y 1 · ∣∣ ∣ ∣ Yky 1| · · · |j Y kxx| · · · |Y ky m ∣ ∣ ∣ ∣1 −(5.18) I :=− y j x X y j · ∣∣ ∣ ∣ Yky j| · · · |j Y kxx | · · · |Y ky m ∣ ∣∣ ∣ − · · · −− X xx · ∣∣ ∣ ∣ Yky 1| · · · | j Y kx | · · · |Y ky m ∣ ∣ ∣ ∣ −− yx j X xx · ∣∣ ∣ Yky 1| · · · ∣ |j Y k kyj| · · · |Y y − · · · −m− X y m · ∣∣ ∣ Yky 1| · · · |j Yxx| k · · · |Yxk ∣ −− y m x X y m · ∣∣ ∣ ∣ Yky 1| · · · |j Y kxx | · · · |Y ky m ∣ ∣∣ ∣1 −− y j x X y m · ∣∣ ∣ ∣ Yky 1| · · · |j Y kxx| · · · |Y ky j ∣ ∣ ∣ ∣ ,

86and secondly (we discuss afterwards the annihilation of the underlinedterms):(5.19) II :=2m∑l 1 =1− 2− 2− 2− 2− 2− 2− 2− 2m∑l=0m∑l 1 =1m∑l 1 =1m∑l 1 =1m∑l 1 =1m∑l 1 =1m∑l 1 =1m∑l 1 =1m∑l 1 =1y l 1x y l x X y l ·y l 1x X y 1 ·∣ ∣∣∣Y ky 1| · · · |j Y kxy l 1| · · · |Yy k ∣∣ ∣∣− m2∣ ∣∣∣Y x k | · · · |j Y kxy 1| · · · |Y k ∣∣ ∣∣ l y − my l 1x y 1 x X y 1 · ∣ ∣∣∣ ∣∣Y ky 1| · · · |j Y kxy l 1 | · · · |Y ky m ∣ ∣∣∣ ∣∣2−y l 1x y j x X y 1 ·∣ ∣∣∣Y ky j| · · · |j Y kxy l 1| · · · |Yy k ∣∣ ∣∣ − · · · −my l 1x X xy l 1 · ∣∣ ∣ ∣ Yky 1| · · · |j Y kx | · · · |Y ky m ∣ ∣ ∣ ∣ −y l 1x yx j X xy l 1 · ∣∣ ∣ Yky 1| · · · ∣ |j Y k kj| · · · |Y ∣y − · · · −my l 1x X y m ·∣∣∣Y ky 1| · · · |j Y kxy | · · · |Y kl 1 x∣ −y l 1x y m x X y m · ∣ ∣∣∣ ∣∣Y ky 1| · · · |j Y kxy l 1 | · · · |Y ky m ∣ ∣∣∣ ∣∣2−y l 1x y j x X y m ·y∣ ∣∣∣Y ky 1| · · · |j Y kxy l 1| · · · |Y k ∣∣ ∣∣y , j

87and where thirdly (we are nearly the end of the proof):(5.20)m∑ m∑ m∑ ∣ ∣ y l 1x y l 2x yx l ∣∣ ∣∣Y X y · kl y 1| · · · |j Y ky l 1y | · · · |Y k ∣∣ ∣∣ l 2 y − m3III :=l 1 =1 l 2 =1 l=0−−−−−−−m∑m∑l 1 =1 l 2 =1m∑m∑l 1 =1 l 2 =1m∑m∑l 1 =1 l 2 =1m∑m∑l 1 =1 l 2 =1m∑m∑l 1 =1 l 2 =1m∑m∑l 1 =1 l 2 =1m∑m∑l 1 =1 l 2 =1y l 1x y l 2x X y 1 ·∣ ∣∣Yx k | · · · |j Y ky l 1y | · · · |Y k ∣∣ ∣∣ l 2 y − my l 1x y l 2x y 1 x X y 1 · ∣ ∣∣∣ ∣∣Y ky 1| · · · |j Y ky l 1y l 2 | · · · |Y ky m ∣ ∣∣∣ ∣∣3−y l 1x y l 2x y j x X y 1 ·∣ ∣∣Y ky j| · · · |j Y ky l 1y l 2| · · · |Yy k ∣∣ ∣∣ − · · · −my l 1x y l 2x X y l 1y l 2 · ∣∣ ∣ ∣ Yky 1| · · · |j Y kx | · · · |Y ky m ∣ ∣∣ ∣ −y l 1x y l 2x yx j X y l 1y l 2 · ∣∣ ∣ Yky 1| · · · ∣ |j Y k kj| · · · |Y ∣y − · · · −my l 1x y l 2x X y m ·∣∣∣Y ky 1| · · · |j Y ky l 1y l 2| · · · |Yxk∣ −y l 1x y l 2x y m x X y m · ∣ ∣∣∣ ∣∣Y ky 1| · · · |j Y ky l 1y l 2 | · · · |Y ky m ∣ ∣∣∣ ∣∣3−y−m∑l 1 =1m∑l 2 =1y l 1x y l 2x y j x X y m · ∣ ∣∣∣ ∣∣Y ky 1| · · · |j Y ky l 1y l 2 | · · · |Y ky j ∣ ∣∣∣ ∣∣ .Now, we explain the annihilation of the underlined terms. Consider I: in thefirst sum ∑ ml=0, all the terms except only the two corresponding to l = 0and to l = j are annihilated by the other terms with 1 appended: indeed,one must take account of the fact that in the expression of I, we have twosums represented by some cdots, the nature of which was defined withoutambiguity in the passage from (5.15) to (5.16).

88Similar simplifications occur for II and for III. Consequently, we obtainfirstly:(5.21) I :=X x · ∣∣ ∣ ∣ Yky 1| · · · |j Y kxx | · · · |Y ky m ∣ ∣ ∣ ∣ ++ y j x X y j · ∣∣ ∣ ∣ Yky 1| · · · |j Y kxx | · · · |Y ky m ∣ ∣∣ ∣ −− X y 1 · ∣∣ ∣ ∣ Ykx | · · · | j Y kxx| · · · |Y ky m ∣ ∣ ∣ ∣ −− yx j X y j · ∣∣ ∣ Yky j| · · · |j Yxx k | · · · |Y ∣ y k∣ − · · · −m− X xx · ∣∣ ∣ ∣Y ky 1| · · · |j Yx k | · · · |Y ∣ y k− m− yx j X xx · ∣∣ ∣ Yky 1| · · · ∣ |j Y k kyj| · · · |Y y − · · · −m− X y m · ∣∣ ∣ Yky 1| · · · |j Yxx k | · · · |Y xk ∣ −− y j x X y m · ∣∣ ∣ ∣ Yky 1| · · · |j Y kxx | · · · |Y ky j ∣ ∣ ∣ ∣ ;just above, the first two lines consist of the two terms in the sum underlinedat the first line of (5.18) which are not annihilated; secondly we obtain:(5.22) II :=2m∑l 1 =1+ 2− 2− 2− 2− 2− 2− 2y l 1x X x ·m∑l 1 =1m∑l 1 =1m∑l 1 =1m∑l 1 =1m∑l 1 =1m∑l 1 =1m∑l 1 =1∣ ∣∣Y ky 1| · · · |j Y kxy | · · · |Y k ∣∣ ∣∣ l 1 y + my l 1x y j x X y j · ∣ ∣∣∣ ∣∣Y ky 1| · · · |j Y kxy l 1 | · · · |Y ky m ∣ ∣∣∣ ∣∣ −y l 1x X y 1 ·y l 1x y j x X y 1 ·∣ ∣∣Yx k | · · · |j Y kxy | · · · |Y k ∣∣ ∣∣ l 1 y − m∣ ∣∣Y ky j| · · · |j Y kxy l 1| · · · |Yy k ∣∣ ∣∣ − · · · −my l 1x X xy l 1 · ∣∣ ∣ ∣ Yky 1| · · · |j Y kx | · · · |Y ky m ∣ ∣ ∣ ∣ −y l 1x y j x X xy l 1 · ∣∣ ∣ ∣ Yky 1| · · · |j Y kx | · · · |Y ky m ∣ ∣∣ ∣ − · · · −y l 1x X y m ·∣∣Y ky 1| · · · |j Y kxy l 1| · · · |Yxk∣ −y l 1x y j x X y m · ∣ ∣∣∣ ∣∣Y ky 1| · · · |j Y kxy l 1 | · · · |Y ky j ∣ ∣∣∣ ∣∣;similarly, the first two lines above consist of the two terms in the sum underlinedat the first line of (5.19) which are not annihilated; and thirdly we

89obtain:(5.23)III :=m∑m∑l 1 =1 l 2 =1+−−−−m∑l 1 =1 l 2 =1m∑y l 1x y l 2x X x ·m∑m∑l 1 =1 l 2 =1m∑m∑l 1 =1 l 2 =1m∑m∑l 1 =1 l 2 =1m∑m∑l 1 =1 l 2 =1∣ ∣∣∣Y ky 1| · · · |j Y ky l 1y | · · · |Y k ∣∣ ∣∣+l 2 y my l 1x y l 2x y j x X y j · ∣ ∣∣∣ ∣∣Y ky 1| · · · |j Y ky l 1y l 2 | · · · |Y ky m ∣ ∣∣∣ ∣∣ −y l 1x y l 2x X y 1 ·y l 1x y l 2x y j x X y 1 ·∣ ∣∣Yx k | · · · |j Y ky l 1y | · · · |Y k ∣∣ ∣∣ l 2 y − m∣ ∣∣Y ky j| · · · |j Y ky l 1y l 2| · · · |Yy k ∣∣ ∣∣ − · · · −my l 1x y l 2x X y l 1y l 2 · ∣∣ ∣ ∣ Yky 1| · · · |j Y kx | · · · |Y ky m ∣ ∣ ∣ ∣ −y l 1x y l 2x yx j X y l 1y l 2 · ∣∣ ∣ Yky 1| · · · ∣ |j Y k kj| · · · |Y ∣y − · · · −my−−m∑m∑l 1 =1 l 2 =1m∑m∑l 1 =1 l 2 =1y l 1x y l 2x X y m ·y l 1x y l 2x y j x X y m ·∣∣∣Y ky 1| · · · |j Y ky l 1y | · · · |Y kl 2 x∣ −∣ ∣∣∣Y ky 1| · · · |j Y ky l 1y l 2| · · · |Y k ∣∣ ∣∣y . jCollecting the odd lines of (5.21), we obtain exactly (m + 1) terms whichcorrespond to the development of the determinant ∆(x| · · · | j xx| · · · |y m )along its first line, modulo permutations of columns of the associatedm × m minors; collecting the even lines of (5.21), we obtain exactly(m + 1) terms which correspond to the development of the determinant−y j x·∆(xx|y1 | · · · |y m ) along its first lines, modulo permutations of columnsof the associated m × m minors. Similar observations hold about II and III.

90In , we may rewrite the final expressions of these three terms: firstly⎧I = ∆(x| · · · | j xx| · · · |y m ) − yx j · ∆(xx|y1 | · · · |y m ),m∑II = 2 y l 1x · ∆(x| · · · | j xy l 1| · · · |y m )−(5.24)⎪⎨⎪⎩III =l 1 =1m∑− 2 y j xm∑l 1 =1 l 2 =1− y j xm∑l 1 =1y l 1x · ∆(xy l 1|y 1 | · · · |y m ),y l 1x y l 2x · ∆(x| · · · | j y l 1y l 2| · · · |y m )−m∑m∑l 1 =1 l 2 =1y l 1x y l 2x · ∆(y l 1y l 2|y 1 | · · · |y m ).Coming back to (5.17), we obtain the desired expression (5.4).The proof of the — technical, though involving only linear algebra —Lemma 3.32 is complete.REFERENCES[BK1989] BLUMAN, G.W.; KUMEI, S.: Symmetries and differential equations,Springer-Verlag, New-York, 1989.[Ca1924] CARTAN, É.: Sur les variétés à connexion projective, Bull. Soc. Math. France52 (1924), 205–241.[Ch1939] CHERN, S.-S.: Sur la géométrie d’un système d’équations différentielles dusecond ordre, Bull. Sci. Math. 63 (1939), 206–212.[CMS1996] CRAMPIN, M.; MARTÍNEZ, E.; SARLET, W.: Linear connections for systemsof second-order ordinary differential equations, Ann. Inst. H. Poincaré Phys.Théor. 65 (1996), no. 2, 223–249.[Do2000] DOUBROV, B.: Contact invariants of ordinary differential equations, RIMSKokyuroku 1150 (2000), 105–113.[DNP2005] DRIDI, R.; NEUT, S.; PETITOT, M.: Élie Cartan’s geometrical vision or howto avoid expression swell, arxiv.org/abs/math.DG/0504203.[Fe1995] FELS, M.: The equivalence problem for systems of second-order ordinarydifferential equations, Proc. London Math. Soc. 71 (1995), no. 2, 221–240.[G1989] GARDNER, R.B.: The method of equivalence and its applications, CBMS-NSF Regional Conference Series in Applied Mathematics 58 (SIAM,Philadelphia, 1989), 127 pp.[GM2003] GAUSSIER, H.; MERKER, J.: Symmetries of partial differential equations, J.Korean Math. Soc. 40 (2003), no. 3, 517–561.[GG1983] GONZÁLEZ GASCÓN, F.; GONZÁLEZ LÓPEZ, A.: Symmetries of differential[GL1988]equations, IV. J. Math. Phys. 24 (1983), 2006–2021.GONZÁLEZ LÓPEZ, A.: Symmetries of linear systems of second order differentialequations, J. Math. Phys. 29 (1988), 1097–1105.[GKO1992] GONZÁLEZ LÓPEZ, A.; KAMRAN, N.; OLVER, P.J.: Lie algebras of vectorfields in the real plane, Proc. London Math. Soc. 64 (1992), 339–368.

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93Nonalgebraizable real analytic tubes in C n xJoël Merker (with H. Gaussier)Abstract. We give necessary conditions for certain real analytic tube generic submanifoldsin C n to be locally algebraizable. As an application, we exhibit familiesof real analytic non locally algebraizable tube generic submanifolds in C n .During the proof, we show that the local CR automorphism group of a minimal,finitely nondegenerate real algebraic generic submanifold is a real algebraic localLie group. We may state one of the main results as follows. Let M be a real analytichypersurface tube in C n passing through the origin, having a defining equation ofthe form v = ϕ(y), where (z,w) = (x+iy,u+iv) ∈ C n−1 ×C. Assume that M isLevi nondegenerate at the origin and that the real Lie algebra of local infinitesimalCR automorphisms of M is of minimal possible dimension n, i.e. generated by thereal parts of the holomorphic vector fields ∂ z1 ,... ,∂ zn−1 ,∂ w . Then M is locallyalgebraizable only if every second derivative ∂ 2 y k y lϕ is an algebraic function of thecollection of first derivatives ∂ y1 ϕ,...,∂ ym ϕ.Table of contents :§1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.§2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.§3. Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.§4. Local Lie group structure for the CR automorphism group . . . . . . . .17.§5. Minimality and finite nondegeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.§6. Algebraicity of local CR automorphism groups . . . . . . . . . . . . . . . . . . . 22.§7. Description of explicit families of strong tubes in C n . . . . . . . . . . . . . . 28.§8. Analyticity versus algebraicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.Mathematische Zeitschrift 247 (2004), no. 2, 337–383§1. INTRODUCTIONA real analytic submanifold M in C n is called algebraic if it can be representedlocally by the vanishing of a collection of Nash algebraic real analyticfunctions. We say that M is locally algebraizable at one of its points p ifthere exist some local holomorphic coordinates centered at p in which Mis algebraic. For instance, every totally real, real analytic submanifold inC n of dimension k ≤ n is locally biholomorphic to a k-dimensional linearreal plane, hence locally algebraizable. Also, every complex manifoldis locally algebraizable. Although every real analytic submanifold M isclearly locally equivalent to its tangent plane by a real analytic (in general

94not holomorphic) equivalence, the question whether M is biholomorphicallyequivalent to a real algebraic submanifold is subtle. In this article,we study the question whether every real analytic CR submanifold is locallyalgebraizable. One of the interests of algebraizability lies in the reflectionprinciple, which is better understood in the algebraic category. Indeed,in the fundamental works of Pinchuk [Pi1975], [Pi1978] and of Webster[We1977], [We1978] and in the recent works of Sharipov-Sukhov [SS1996],Huang-Ji [HJ1998], Verma [Ve1999], Coupet-Pinchuk-Sukhov [CPS2000],and Shafikov [Sha2000], [Sha2002], the extendability of germs of CR mappingswith target in a real algebraic hypersurface is achieved. On the contrary,even if some results previously shown under an algebraization hypothesiswere proved recently under general assumptions (see the strong resultobtained by Diederich-Pinchuk [DP2003]), most of the results cited aboveare still open in the case of a real analytic target hypersurface.1.1. Brief history of the question. By the work of Moser and Webster[MW1983, Thm. 1], it is known that every real analytic two-dimensionalsurface S ⊂ C 2 at an isolated elliptic (in the sense of Bishop) complex tangencyp ∈ S is biholomorphic to one of the surfaces S γ,δ,s := {(z 1 , z 2 ) ∈C 2 : y 2 = 0, x 2 = z 1¯z 1 + (γ + δ(x 2 ) s )(z1 2 + ¯z2 1 )}, where p corresponds tothe origin, where 0 < γ < 1/2 is Bishop’s invariant and where δ = ±1 ands ∈ N or δ = 0. The quantities γ, δ, s form a complete system of biholomorphicinvariants for the surface S near p. In particular, every elliptic surfaceS ⊂ C 2 is locally algebraizable. To the authors’ knowledge, it is unknownwhether there exist nonalgebraizable hyperbolic surfaces in C 2 . In fact, veryfew examples of nonalgebraizable submanifolds are known. In [Eb1996],the author constructed a nonminimal (and non Levi-flat) real analytic hypersurfaceM through the origin in C 2 which is not locally algebraizable (cf.[BER2000, p. 330]). In a recent article [HJY] the authors prove that thestrongly pseudoconvex real analytic hypersurface Im w = e |z|2 − 1 passingthrough the origin in C 2 is not locally algebraizable at any of its points. Usingan associated projective structure bundle Y introduced by Chern, theyshow that for every rigid algebraic hypersurface in C n , there exists an algebraicdependence relation between seven explicit Cartan-type holomorphicinvariant functions on Y . However a computational approach shows thatwhen M is of the specific form Im w = e |z|2 − 1, no algebraic relation canbe satisfied by these seven invariants.1.2. Presentation of the main results. Our aim is to present a geometricalapproach of the problem, valid in arbitrary dimension and in arbitrarycodimension, and to exhibit a large class of nonalgebraizable real analyticgeneric submanifolds. We consider the class Tn d of generic real analyticsubmanifolds in C n passing through the origin, of codimension d ≥ 1

and of CR dimension m = n − d ≥ 1, whose local CR automorphismgroup is n-dimensional, generated by the real parts of n holomorphic vectorfields having holomorphic coefficients X 1 , . . .,X n which are linearlyindependent at the origin and which commute: [X i1 , X i2 ] = 0. We shallcall Tn d the class of strong tubes of codimension d. Indeed, since thereexists a straightened system of coordinates t = (t 1 , . . .,t n ) over C n inwhich X i = ∂ ti , we observe that every submanifold M ∈ Tn d is tubifiableat the origin. By this, we mean that there exist holomorphic coordinatest = (z, w) = (x + iy, u + iv) ∈ C m × C d vanishing at the origin in whichM is represented by d equations of the form v j = ϕ j (y). Hence M is a tube,i.e. a product of the submanifold {v j = ϕ j (y), j = 1, . . .,d} ⊂ R n y,vby the n-dimensional real space R n x,u . Since M ∈ T n d , the only infinitesimalCR automorphisms of M are the real parts of the vector fields∂ z1 , . . .,∂ zm , ∂ w1 , . . .,∂ wd , explaining the terminology. Notice that not everytube belongs to the class Tn d . For instance in codimension d = 1, theHeisenberg sphere v = ∑ n−1k=1 y2 k and more generally the Levi nondegeneratequadrics v = ∑ n−1k=1 ε k yk 2, where ε k = ±1, have a CR automorphismgroup of dimension (n+1) 2 −1 > n and so do not belong to Tn 1 . We assumethat M ∈ Tn d is minimal at the origin, namely the local CR orbit of 0 in Mcontains a neighborhood of 0 in M. Furthermore, we assume that M ∈ Tndis finitely nondegenerate at 0, namely that there exists an integer l ≥ 1 suchthat Span {L β ∇ t (r j )(0, 0) : β ∈ N m , |β| ≤ l, j = 1, . . ., d} = C n , wherer j (t, ¯t) = 0, j = 1, . . .,d are arbitrary real analytic defining functions for Mnear 0 satisfying ∂r 1 ∧· · ·∧∂r d ≠ 0 on M, where ∇ t (r j )(t, ¯t) is the holomorphicgradient with respect to t of r j and where L β denotes (L 1 ) β1 · · ·(L m ) βmfor an arbitrary basis L 1 , . . ., L m of (0, 1)-vector fields tangent to M ina neighborhood of 0. In particular Levi nondegenerate hypersurfaces arefinitely nondegenerate. Finally, assuming only that ϕ j (0) = 0, j = 1, . . ., d,we shall observe in Lemma 3.2 below that a tube v j = ϕ j (y) of codimensiond is finitely nondegenerate at the origin if and only if there exist multi-indicesβ∗ 1, . . .,βm ∗ ∈ Nm with |β∗ k| ≥ 1 and integers 1 ≤ j1 ∗ , . . .,jm ∗ ≤ d such thatthe real mapping(1.1) ψ(y) :=(∂|β 1 ∗ | ϕ j 1 ∗(y)∂y β1 ∗), . . ., ∂|βm ∗ | ϕ j m ∗(y)=: y ′ ∈ R m∂y βm ∗is of rank m at the origin in R m . Our main theorem provides a necessarycondition for the local algebraizability of strong tubes :Theorem 1.1. Let M be a real analytic generic tube of codimension d inC n given in coordinates (z, w) = (x + iy, u + iv) ∈ C m × C d by theequations v j = ϕ j (y), where ϕ j (0) = 0, j = 1, . . ., d. Assume that Mis minimal and finitely nondegenerate at the origin, so the real mapping95

96ψ(y) = y ′ defined by (1.1) is of rank m at the origin in R m y and let y = ψ ′ (y ′ )denote the local inverse in ψ(y). Assume that M ∈ Tn d , namely M is astrong tube of codimension d. If M is locally algebraizable at the origin,then all the derivative functions ∂ y ′kψ l ′(y′ ), where 1 ≤ k, l ≤ m, are realalgebraic functions of y ′ . Equivalently, every second derivative ∂y 2 k y lϕ j (y) isan algebraic function of the collection of first derivatives ∂ y1 ϕ j , . . .,∂ ym ϕ j .By contraposition, every real analytic strong tube M ∈ Tn d for whichone of the derivative functions ∂ y ′kψ l ′ is not real algebraic is not locally algebraizable.We will argue in §8 that this is generically the case in the senseof Baire. It is however natural to look for explicit examples of nonalgebraizablereal analytic submanifolds in C n . Since the real parts of the vectorfields ∂ z1 , . . .,∂ zm , ∂ w1 , . . ., ∂ wd are infinitesimal CR automorphisms of everytube v = ϕ(y), we must provide some sufficient conditions insuringthat the dimension of the Lie algebra of such a tube is exactly n. We shallestablish in §§7-8 below:Corollary 1.2. The tube hypersurface M χ1 ,...,χ n−1in C n of equation v =∑ n−1k=1 [ε kyk 2+y6 k +y9 k y 1 · · ·y k−1 +y n+8kχ k (y 1 , . . .,y n−1 )], where χ 1 , . . .,χ n−1are arbitrary real analytic functions, belongs to the class Tn 1 of strong tubes.Two such tubes M χ1 ,...,χ n−1and M bχ1 ,...,bχ n−1are biholomorphically equivalentif and only if χ j = ̂χ j for every j. Furthermore, for a generic choice inχ 1 , . . .,χ n−1 in the sense of Baire (to be precised in §8), M χ1 ,...,χ n−1is notlocally algebraizable at the origin.Here we annihilate some Taylor coefficients in ϕ and keep some othersto be nonzero to insure that M χ is a strong tube. Furthermore, the termsy 9 k y 1 · · ·y k−1 insure that the M χ are pairwise not biholomorphically equivalent.Using a classical direct algorithm (cf. [Bs1991], [St1991]), or the Lietheory of symmetries of differential equations, combined with Theorem 1.1we may provide some other explicit strong tubes which are not locally algebraizable(see §§7-8 for the proof):Corollary 1.3. The following five explicit tubes belong to T2 1 and are notlocally algebraizable at the origin : v = sin(y 2 ), v = tan(y 2 ), v = e ey−1 −1,v = sinh(y 2 ) and v = tanh(y 2 ).In these five examples, the algebraic independence in ∂ y ϕ and in ∂ 2 yy ϕis clear; however, checking that each hypersurface is indeed a strong tuberequires some formal computations, see §7. One may also check by a directcomputation that in a neighborhood of every point p = (z p , w p ) withz p ≠ 0, the hypersurface M HJY of global equation Im w = e |z|2 − 1 is astrong tube (see §7.5). Since it can be represented in a neighborhood of punder the tube form v ′ = e |zp|2 (e y′ −1) − 1 by means of the local change of

coordinates z ′ = 2i ln(z/z p ), w ′ = (w − w p ) e −|zp|2 , applying Theorem 1.1and inspecting the function e |zp|2 (e y′ −1) −1, we may check that it is not algebraizableat such points p with z p ≠ 0 (see §7.5). It follows trivially that thehypersurface M HJY is also not locally algebraizable at all the points p withz p = 0, giving the result of [HJY, Theorem 1.1]. Using the same strategy asfor Theorem 1.1, we obtain more generally the following criterion:Theorem 1.4. Let M ϕ : v = ϕ(z¯z) be a Levi nondegenerate real analytichypersurface in C 2 passing through the origin whose Lie algebra of localinfinitesimal CR automorphisms is generated by ∂ w and iz ∂ z . If M ϕ is locallyalgebraizable at the origin, then the first derivative ∂ r ϕ in ϕ (r ∈ R)is algebraic. For instance, the following seven explicit examples are not locallyalgebraizable at the origin : v = e z¯z − 1, v = sin(z¯z), v = tan(z¯z),v = sinh(z¯z), v = tanh(z¯z), v = sin(sin(z¯z)) and v = e ez¯z −1 − 1.Finally, using the same recipe as for Theorems 1.1 and 1.4, we shall providea very simple criterion for the local nonalgebraizability of some hypersurfaceshaving a local Lie CR automorphism group of dimension equal toone exactly. We consider the class R n of Levi nondegenerate real analytichypersurfaces passing through the origin in C n (n ≥ 2) such that the Lie algebraof infinitesimal CR automorphisms of M is generated by exactly oneholomorphic vector field X 1 with holomorphic coefficients not all vanishingat the origin. We call R n the class of strongly rigid hypersurfaces, in orderto distinguish them from the so-called rigid ones whose local CR automorphismgroup may be of dimension larger than 1. By straightening X 1 , wemay assume that X 1 = ∂ w and that M is given by a real analytic equationof the form v = ϕ(z, ¯z) = ϕ(z 1 , . . .,z n−1 , ¯z 1 , . . ., ¯z n−1 ). By making someelementary changes of coordinates (cf. §3.3), we can furthermore assumewithout loss of generality that ϕ(z, ¯z) = ∑ n−1k=1 ε k |z k | 2 + χ(z, ¯z), whereε k = ±1 and χ(0, ¯z) ≡ ∂ zk χ(0, ¯z) ≡ 0.Theorem 1.5. Let M : v = ϕ(z, ¯z) = ∑ n−1k=1 ε k |z k | 2 +χ(z, ¯z) be a stronglyrigid hypersurface in C n with χ(0, ¯z) ≡ ∂ zk χ(0, ¯z) ≡ 0. If M is locallyalgebraizable at the origin, then all the first derivatives ∂ zk ϕ are algebraicfunctions of (z, ¯z).This criterion enables us to exhibit a whole family of non locally algebraizablehypersurfaces in C n :Corollary 1.6. The rigid hypersurfaces M χ1 ,...,χ n−1in C n of equation v =∑ n−1k=1 [ε k |z k | 2 + |z k | 10 + |z k | 14 + |z k | 16 (z k + ¯z k ) + |z k | 18 |z 1 | 2 · · · |z k−1 | 2 +|z k | 2n+16 χ k (z, ¯z)], where the χ k are arbitrary real analytic functions, belongto the class R n of strongly rigid hypersurfaces. Two such tubesM χ1 ,...,χ n−1and M bχ1 ,...,bχ n−1are biholomorphically equivalent if and only if97

98χ k = ̂χ k for k = 1, . . .,n − 1. Furthermore, for a generic choice of a(n − 1)-tuple of real analytic functions (χ 1 , . . .,χ n−1 ) in the sense of Baire(to be precised in §8), M χ1 ,...,χ n−1is not locally algebraizable at the origin.Finally, by computing generators of the Lie algebra of local infinitesimalCR automorphisms of some explicit examples, we obtain:Corollary 1.7. The following seven explicit examples of hypersurfaces inC 2 are strongly rigid and are not locally algebraizable at the origin : v =z¯z + z 2¯z 2 sin(z + ¯z), v = z¯z + z 2¯z 2 exp(z + ¯z), v = z¯z + z 2¯z 2 cos(z + ¯z),v = z¯z+z 2¯z 2 tan(z+¯z), v = z¯z+z 2¯z 2 sinh(z+¯z), v = z¯z+z 2¯z 2 cosh(z+¯z)and v = z¯z + z 2¯z 2 tanh(z + ¯z).1.3. Content of the paper. To prove Theorem 1.1 we consider an algebraicequivalent M ′ of M. The main technical part of the proof consists in showingthat an arbitrary real algebraic element M ′ of Tn d can be straightened insome local complex algebraic coordinates t ′ ∈ C n in order that its infinitesimalCR automorphisms are the real parts of n holomorphic vector fields ofthe form X i ′ = c′ i (t′ i ) ∂ t ′ , i = 1, . . .,n, where the variables are separated andithe functions c ′ i (t′ i ) are algebraic. For this, we need to show that the automorphismgroup of a minimal finitely nondegenerate real algebraic genericsubmanifold in C n is a local real algebraic Lie group, a notion defined in§2.3. A large part of this article (§§4, 5, 6) is devoted to provide an explicitrepresentation formula for the local biholomorphic self-transformations ofa minimal finitely nondegenerate generic submanifold, see especially Theorem2.1 and Theorem 4.1. Finally, using the specific simplified form ofthe vector fields X i ′ and assuming that there exists a biholomorphic equivalenceΦ : M → M ′ satisfying Φ ∗ (∂ ti ) = c ′ i (t′ i ) ∂ t ′ , we show by elementaryicomputations that all the first order derivatives of the mapping ψ ′ (y ′ ) mustbe algebraic. We follow a similar strategy for the proofs of Theorems 1.4and 1.5. Finally, in §§7-8, we provide the proofs of Corollaries 1.2, 1.3, 1.6and 1.7.1.4. Acknowledgment. We acknowledge interesting discussions withMichel Petitot François Boulier at the University of Lille 1.§2. PRELIMINARIESWe recall in this section the basic properties of the objects we will dealwith.2.1. Nash algebraic functions and manifolds. In this subsection, let K =R or C. Let (x 1 , . . .,x n ) denote coordinates over K n . Throughout thearticle, we shall use the norm |x| := max(|x 1 |, . . .,|x n |) for x ∈ K n .Let K be an open polydisc centered at the origin in K n , namely K =

{x ∈ K n : |x| < ρ} for some ρ > 0. Let f : K → K be a K-analytic function, defined by a power series converging normally in K .We say that f is (Nash) K-algebraic if there exists a nonzero polynomialP(X 1 , . . ., X n , F) ∈ K[X 1 , . . .,X n , F] in (n + 1) variables such that therelation P(x 1 , . . .,x n , f(x 1 , . . .,x n )) = 0 holds for all (x 1 , . . .,x n ) ∈ K .If K = R, we say that f is real algebraic. If K = C, we say that f is complexalgebraic. The category of K-algebraic functions is stable under elementaryalgebraic operations, under differentiation and under composition.Furthermore, implicit solutions of K-algebraic equations (for which the realanalytic implicit function theorem applies) are again K-algebraic mappings.The theory of K-algebraic manifolds is then defined by the usual axioms ofmanifolds, for which the authorized changes of chart are K-algebraic mappingsonly (cf. [Za1995]). In this paper, we shall very often use the stabilityof algebraicity under differentiation.2.2. Infinitesimal CR automorphisms. Let M ⊂ C n be a generic submanifoldof codimension d ≥ 1 and CR dimension m = n − d ≥ 1. Let p ∈ M,let t = (t 1 , . . ., t n ) be some holomorphic coordinates vanishing at p and forsome ρ > 0, let ∆ n (ρ) := {t ∈ C n : |t| < ρ} be an open polydisc centeredat p. We consider the Lie algebra Hol(∆ n (ρ)) of holomorphic vector fieldsof the form X = ∑ nj=1 a j(t) ∂/∂t j , where the a j are holomorphic functionsin ∆ n (ρ). Here, Hol(∆ n (ρ)) is equipped with the usual Jacobi-Lie bracketoperation. We may consider the complex flow exp(σX)(q) of a vector fieldX ∈ Hol(∆ n (ρ)). It is a holomorphic map of the variables (σ, q) whichis well defined in some connected open neighborhood of {0} × ∆ n (ρ) inC × ∆ n (ρ).Let K denote the real vector field K := X + X, considered as a realvector field over R 2n ∼ = C n . Again, the real flow of K is defined in someconnected open neighborhood of {0} × ∆ n (ρ) R in R × ∆ n (ρ) R . We remindthe following elementary relation between the flow of K and the flow of X.For a real time parameter σ := s ∈ R, the flow exp(sX)(q) coincides withthe real flow of X + X, namely exp(sX)(q) = exp(s(X + X))(q R ), wherefor q ∈ C n , we denote q R the corresponding real point in R 2n . In the sequel,we shall always identify ∆ n (ρ) and its real counterpart ∆ n (ρ) R .Let now Hol(M, ∆ n (ρ)) denote the real subalgebra of the vectorfields X ∈ Hol(∆ n (ρ)) such that X + X is tangent to M ∩ ∆ n (ρ).We also denote by Aut CR (M, ∆ n (ρ)) the Lie algebra of vectorfields of the form X + X, where X belongs to Hol(M, ∆ n (ρ)), soAut CR (M, ∆ n (ρ)) = 2 ReHol(M, ∆ n (ρ)). By the above considerations,the local flow exp(sX)(q) of X with s ∈ R real makes a one-parameterfamily of local biholomorphic transformations of M. In the sequel, we shallalways identify Hol(M, ∆ n (ρ)) and Aut CR (M, ∆ n (ρ)), namely we shall99

100identify X and X + X and say by some abuse of language that X itself isan infinitesimal CR automorphism.In the algebraic category, the main drawback of infinitesimal CR automorphismis that they do not have algebraic flow. For instance, the complexdilatation vector field X = iz ∂ z has transcendent flow, even if it is an infinitesimalCR automorphism of every algebraic hypersurface in C 2 whoseequation is of the form v = ϕ(z¯z), even if the coefficient of X is algebraic.Thus instead of infinitesimal CR automorphisms which generate oneparametergroups of biholomorphic transformations of M, we shall studyalgebraically dependent one-parameter families of biholomorphic transformations(not necessarily making a one parameter group). To begin with, weneed to introduce some precise definitions about local algebraic Lie transformationgroups.2.3. Local Lie group actions in the K-algebraic category. Often in real orin complex analytic geometry, the interest cannot be focalized on global Lietransformation groups, but only on local transformations which are close tothe identity. For instance, the transformation group of a small piece of a realanalytic CR manifold in C n which is not contained in a global, large or compactCR manifold is almost never a true, global transformation group. Consequentlythe usual axioms of Lie transformation groups must be localized.Philosophically speaking, the local point of view is often the most adequateand the richest one, because a given analytico-geometric object often possessesmuch more local invariant than global invariants, if any. Historicallyspeaking, the local Lie transformation groups were first studied, before theintroduction of the now classical notion of global Lie group. Especially, inhis first masterpiece work [Lie1880] on the subject, Sophus Lie essentiallydealt with local “Lie” groups: he classified all continuous local transformationgroups acting on an open subset of C 2 . This general classificationprovided afterwards in the years 1880–1890 many applications to the localstudy of differential equations: local normal forms, local solvability, etc.In this paragraph we define precisely local actions of local Lie groups andwe focus especially on the K-algebraic category.Let c ∈ N ∗ , let g = (g 1 , . . .,g c ) ∈ K c and let two positive numberssatisfy 0 < δ 2 < δ 1 . We formulate the desired definition by means of thetwo precise polydiscs ∆ c (δ 2 ) ⊂ ∆ c (δ 1 ) ⊂ K c . A local K-algebraic Liegroup of dimension c consists of the following data:(1) A K-algebraic multiplication mapping µ : ∆ c (δ 2 ) × ∆ c (δ 2 ) → ∆ c (δ 1 )which is locally associative (µ(g, µ(g ′ , g ′′ )) = µ(µ(g, g ′ ), g ′′ )), wheneverµ(g ′ , g ′′ ) ∈ ∆ c (δ 2 ), µ(g, g ′ ) ∈ ∆ c (δ 2 ) and which satisfiesµ(0, g) = µ(g, 0) = g, where the origin 0 ∈ K c corresponds to theidentity element in the group structure.

(2) A K-algebraic inversion mapping ι : ∆ c (δ 2 ) → ∆ c (δ 1 ) satisfyingµ(g, ι(g)) = µ(ι(g), g) = 0 and ι(0) = 0 whenever ι(g) ∈ ∆ c (δ 2 ).Here, the integer c ∈ N ∗ is the dimension of G. We shall say that compositionand inversion are defined locally in a neighborhood of the identityelement. In the K-analytic category, the corresponding definition is similar.Now, we can define the notion of local K-algebraic Lie group action. Letn ∈ N ∗ , let x = (x 1 , . . ., x n ) ∈ K n and let two positive numbers satisfy 0

102Indeed, from the group property Φ(Φ(z, w; σ); σ ′ ) ≡ Φ(z, w; σ + σ ′ ), it isclassical and immediate to deduce that if we define the parameter independentvector field X 0 (z, w) := ∂ σ Φ(z, w; σ)| σ=0 = z ∂ z , then it holds that∂ σ Φ(z, w; σ) = e σ z ∂ z = X 0 (Φ(z, w; σ)). So the infinitesimal generatorof the action is independent of the parameter g. However, the main troublehere is that the algebraicity of the action is necessarily lost since the flow ofX 0 is not algebraic (the reader may check that each right (or left) invariantvector field on an algebraic local Lie group defines in general a nonalgebraicone-parameter subgroup, e.g. for SO(2, R), SL(2, C)).Consequently we may allow the infinitesimal generators of an algebraiclocal Lie group action x ′ = Φ(x; g), defined by X i (x; g i ) :=[∂ gi Φ i ](Φ −1i,g i(x); g i ) to depend on the group parameter g i , even if the families(Φ i,gi (x)) gi ∈K do not constitute one-dimensional subgroups of transformations.2.4. Algebraicity of complex flow foliations. Suppose now that M is a realalgebraic generic submanifold in C n , for instance a hypersurface which isLevi nondegenerate at a “center” point p ∈ M corresponding to the originin the coordinates t = (t 1 , . . .,t n ). Let X ∈ Hol(M) be an infinitesimalCR automorphism. Even if, for fixed real s, the biholomorphicmapping t ↦→ exp(sX)(t) is complex algebraic, i.e. the n components ofthis biholomorphism are complex algebraic functions by Webster’s theorem[We1977], we know by considering the infinitesimal CR automorphismX 1 := i(z + 1)∂ z of the strong tube Im w = |z + 1| 2 + |z + 1| 6 − 2 in C 2passing through the origin, that the flow of X is not necessarily algebraicwith respect to all variables (s, t).Nevertheless, we shall show that the local CR automorphism group ofM is a local algebraic Lie group whose general transformations are of theform t ′ = H(t; e 1 , . . .,e c ), where t ∈ C n and (e 1 , . . ., e c ) ∈ R c and whereH is algebraic with respect to all its variables. Thus the “time” dependentvector fields defined by X i (t; e i ) := [∂ ei H i ](H −1i,e i(t); e i ), where H i,ei (t) :=H i (t; e i ) := H(t; 0, . . ., 0, e i , 0, . . ., 0), have an algebraic flow, simply givenby (t, e i ) ↦→ H i (t; e i ). It follows that each foliation defined by the complexintegral curves of the time dependent complex vector fields X i , i = 1, . . .,c,is a complex algebraic foliation, see §3 below. Now, we can state the maintechnical theorem of this paper, whose proof is postponed to §4, §5 and §6.Theorem 2.1. Let M ⊂ C n be a real algebraic connected geometricallysmooth generic submanifold of codimension d ≥ 1 and CR dimension m =n − d ≥ 1. Let p ∈ M and assume that M is finitely nondegenerate andminimal at p. Then for every sufficiently small nonempty open polydisc ∆ 1centered at p, the following three properties hold:

(1) The complex Lie algebra Hol(M, ∆ 1 ) is of finite dimension c ∈ Nwhich depends only on the local geometry of M in a neighborhood ofp.(2) There exists a nonempty open polydisc ∆ 2 ⊂ ∆ 1 also centered at pand a C n -valued mapping H(t; e) = H(t; e 1 , . . .,e c ) with H(t; 0) ≡ twhich is defined in a neighborhood of the origin in C n × R c and whichis algebraic with respect to both its variables t ∈ C n and e ∈ R c suchthat for every holomorphic map h : ∆ 2 → ∆ 1 with h(∆ 2 ∩ M) ⊂∆ 1 ∩ M which is sufficiently close to the identity map, there exists aunique e ∈ R c such that h(t) = H(t; e).(3) The mapping (t, e) ↦→ H(t; e) constitutes a K-algebraic local Lietransformation group action. More precisely, there exist a local multiplicationmapping (e, e ′ ) ↦→ µ(e, e ′ ) and a local inversion mappinge ↦→ ι(e) such that H, µ and ι satisfy the axioms of local algebraic Liegroup action as defined in §2.3.(4) The c “time dependent” holomorphic vector fields(2.1) X i (t; e i ) := [∂ ei H i ](H −1i,e i(t); e i ),103where H i,ei (t) := H i (t; e i ) := H(t; 0, . . ., 0, e i , 0, . . ., 0), have algebraiccoefficients and have an algebraic flow, given by (t, e i ) ↦→H i (t; e i ).In the case where M is real analytic, the same theorem holds true with theword “algebraic” everywhere replaced by the word “analytic”.A special case of Theorem 2.1 was proved in [BER1999b] where, apparently,the authors do not deal with the notion of local Lie groups andconsider the isotropy group of the point p, namely the group of holomorphicself-maps of M fixing p. The consideration of the complete local Lie groupof biholomorphic self-maps of a piece of M in a neighborhood of p (notonly the isotropy group of p) is crucial for our purpose, since we shall haveto deal with strong tubes M ∈ Tn d for which the isotropy group of p ∈ Mis trivial. Sections §4, §5 and §6 are devoted to the proof of Theorem 4.1, aprecise statement of Theorem 2.1. We mention that our method of proof ofTheorem 2.1 gives a non optimal bound for the dimension of Hol(M, ∆ 1 ).To our knowledge, the upper bound c ≤ (n + 1) 2 − 1 is optimal only incodimension d = 1 and in the Levi nondegenerate case.§3. PROOF OF THEOREM 1.1We take in this section Theorem 2.1 for granted. As explained in §1.3above, we shall conduct the proof of Theorem 1.1 in two essential steps(§§3.1 and 3.2). The strategy for the proof of Theorems 1.4 and 1.5 is similarand we prove them in §§3.3 and 3.4. Let M ∈ Tn d be a strong tube of

104codimension d passing through the origin in C n given by the equations v j =ϕ j (y), j = 1, . . .,d. Assume that M is biholomorphically equivalent to areal algebraic generic submanifold M ′ .First step. We show that an arbitrary real algebraic element M ′ ∈ Tn d can bestraightened in some local complex algebraic coordinates t ′ = (t ′ 1 , ..., t′ n ) ∈C n in order that its infinitesimal CR automorphisms are the n holomorphicvector fields of the specific form X i ′ = c ′ i(t ′ i) ∂ t ′i, i = 1, . . ., n, where thefunctions c ′ i (t′ i ) are algebraic.Second step. Assuming that there exists a biholomorphic equivalence Φ :M → M ′ satisfying Φ ∗ (∂ ti ) = c ′ i (t′ ) ∂ t ′i, we prove by direct computationthat all the first order derivatives of the mapping ψ ′ (y ′ ) must be algebraic.3.1. Proof of the first step. Let t ′ = Φ(t) be such an equivalence, withΦ(0) = 0 and M ′ := Φ(M) real algebraic. Let X i := ∂ ti , i = 1, . . .,n,be the n infinitesimal CR automorphisms of M and set X ′ i := Φ ∗ (X i ). Ofcourse, we have [X ′ i 1, X ′ i 2] = Φ ∗ ([X i1 , X i2 ]) = 0, so the CR automorphismgroup of M ′ is also n-dimensional and commutative. Let us choose complexalgebraic coordinates t ′ in a neighborhood of 0 ∈ M ′ such that X ′ i | 0 =∂ t ′i| 0 . Let us apply Theorem 2.1 to the real algebraic submanifold M ′ , notingall the datas with dashes. There exists an algebraic mapping H ′ (t ′ ; e) =H ′ (t ′ ; e 1 , . . .,e n ) such that every local biholomorphic self-map of M ′ writesuniquely t ′ ↦→ H ′ (t ′ ; e), for some e ∈ R n . In particular, for every i =1, . . ., n and every small s ∈ R, there exists e s ∈ R n depending on s suchthat exp(sX ′ i)(t ′ ) ≡ H ′ (t ′ ; e s ). From the commutativity of the flows of theX ′ i , i.e. from exp(s 1X ′ i 1(exp(s 2 X ′ i 2(t ′ )))) ≡ exp(s 2 X ′ i 2(exp(s 1 X ′ i 1(t ′ )))),we get(3.1) H ′ (H ′ (t ′ ; e 2 ); e 1 ) ≡ H ′ (H ′ (t ′ ; e 1 ); e 2 ).This shows that the biholomorphisms t ′ ↦→ H e ′(t′ ) := H ′ (t ′ ; e) commutepairwise. In particular, if we define(3.2) G ′ i (t′ ; e i ) := H ′ (t ′ ; 0, . . .,0, e i , 0, . . .,0),we have G ′ i 1(G ′ i 2(t ′ ; e 2 ); e 1 ) ≡ G ′ i 2(G ′ i 1(t ′ ; e 1 ); e 2 ).Next, after making a linear change of coordinates in the e-space, we caninsure that ∂ ei G ′ i (0; e i)| ei =0 = ∂ t ′i| 0 = X ′ i | 0 for i = 1, . . .,n. Finally, complexifyingthe real variable e i in a complex variable ǫ i , we get mappingsG ′ i (t′ i ; ǫ i) which are complex algebraic with respect to both variables t ′ ∈ C nand ǫ i ∈ C and which commute pairwise. We can now state and prove thefollowing crucial proposition (where we have dropped the dashes) accordingto which we can straighten commonly the n one-parameter families ofbiholomorphisms t ′ ↦→ G ′ i (t′ ; ǫ i ).

Proposition 3.1. Let t ↦→ G i (t; ǫ i ), i = 1, . . ., n, be n one complex parameterfamilies of complex algebraic biholomorphic maps from a neighborhoodof 0 in C n onto a neighborhood of 0 in C n satisfying G i (t; 0) ≡ t,∂ ǫi G i (0; ǫ i )| ǫi =0 = ∂ ti | 0 and pairwise commuting: G i1 (G i2 (t; ǫ 2 ); ǫ 1 ) ≡G i2 (G i1 (t; ǫ 1 ); ǫ 2 ). Then there exists a complex algebraic biholomorphismof the form t ′ ↦→ Φ ′ (t ′ ) =: t of C n fixing the origin with dΦ ′ (0) = Id suchthat if we set G ′ i(t ′ ; ǫ i ) := Φ ′ −1 (G i (Φ ′ (t ′ ); ǫ i )), where t ′ = Φ(t) denote theinverse of t = Φ ′ (t ′ ), then we have(3.3) G ′ i (t′ ; ǫ i ) ≡ (t ′ 1 , . . .,t′ i−1 , G′ i,i (t′ i ; ǫ i), t ′ i+1 , . . .,t′ n ),where the functions G ′ i,i are complex algebraic, depend only on t′ i (and onǫ i ) and satisfy G ′ i,i(t ′ i; 0) ≡ t ′ i and ∂ ǫi G ′ i,i(0; ǫ i )| ǫi =0 = 1.Proof. First of all, we define the complex algebraic biholomorphism(3.4) Φ ′ 1 : (t′ 1 , t′ 2 , . . .,t′ n ) ↦−→ G 1(0, t ′ 2 , . . .,t′ n ; t′ 1 ) =: t.We have dΦ ′ 1(0) = Id, because G 1 (t; 0) ≡ t and ∂ ǫ1 G 1 (0; ǫ 1 )| ǫ1 =0 = ∂ t1 | 0 .Furthermore, since ∂ ǫ1 G 1 (0; ǫ 1 )| ǫ1 =0 is transversal to {(0, t 2 , . . .,t n )}, italso follows that a small neighborhood of the origin in C n t is algebraicallyfoliated by the (n − 1)-parameter family of complex curves C ′t :=′ 2 ,...,t′ n{G 1 (0, t ′ 2, . . .,t ′ n; t ′ 1) : |t ′ 1| < δ} where δ > 0 is small and t ′ 2, . . ., t ′ n arefixed. The existence of this foliation shows that the relation(3.5) t ∗ ∼ t iff there exists ǫ 1 such that t ∗ = G 1 (t; ǫ 1 )is a local equivalence relation, whose equivalence classes are the leaves(see FIGURE 1).C ′t ′ 2 ,...,t′ n105t 2, . . . , t nt ∗ ∼ tC ′ t ′ 2 ,...,t′ nG 1(G 1(0, t ′ 2 , . . . , t′ n ; t′ 1 ); ǫ1)G 1(0, t ′ G 2 , . . . , t′ n ; t′ 1 )1(0, t ′ 2 , . . . , t′ n ; 0)0t 1FIGURE 1: LOCAL ALGEBRAIC STRAIGHTENING OF THE ORBITS OF G 1(t; ǫ 1)Consequently, as we clearly have(3.6) (0, t ′ 2 , . . .,t′ n ) ∼ G 1(0, t ′ 2 , . . .,t′ n ; t′ 1 ) ∼ G 1(G 1 (0, t ′ 2 , . . ., t′ n ; t′ 1 ); ǫ 1),using the transitivity of the relation ∼, it follows that there exists a complexnumber ε t ′ ,ǫ 1depending on t ′ and on ǫ 1 such that(3.7) G 1 (G 1 (0, t ′ 2 , . . ., t′ n ; t′ 1 ); ǫ 1) = G 1 (0, t ′ 2 , . . .,t′ n ; ε t ′ ,ǫ 1).

106By the very definition (3.4) of Φ ′ 1, this is equivalent to(3.8) Φ 1 (G 1 (Φ ′ 1 (t′ ); ǫ 1 )) = (ε t ′ ,ǫ 1, t ′ 2 , . . .,t′ n ),where t ′ = Φ 1 (t) denotes the inverse of t = Φ ′ 1 (t′ ). Finally, since theleft hand side of (3.8) is clearly a complex algebraic mapping of (t ′ ; ǫ 1 ), itfollows that there exists a complex algebraic function G ′ 1,1 (t′ ; ǫ 1 ) such thatwe can write(3.9) Φ 1 (G 1 (Φ ′ 1 (t′ ); ǫ 1 )) ≡ (G ′ 1,1 (t′ ; ǫ 1 ), t ′ 2 , . . .,t′ n ).So we have straightened the first family by means of Φ ′ 1.Next, we drop the dashes and we restart with G 1 (t; ǫ 1 ) =(G 1,1 (t; ǫ 1 ), t 2 , . . .,t n ). Then, similarly as above, by introducing thecomplex algebraic biholomorphism(3.10) Φ ′ 2 : (t ′ 1, t ′ 2, t ′ 3, . . ., t ′ n) ↦−→ G 2 (t ′ 1, 0, t ′ 3, . . .,t ′ n; t ′ 2),which satisfies dΦ ′ 2 (0) = Id, and by denoting by t′ = Φ 2 (t) the inverse oft = Φ ′ 2(t ′ ), we get again that if we set G ′ 2(t ′ ; ǫ 2 ) := Φ 2 (G 2 (Φ ′ 2(t ′ ); ǫ 2 )), then(3.11) G ′ 2 (t′ ; ǫ 2 ) ≡ (t ′ 1 , G′ 2,2 (t′ ; ǫ 2 ), t ′ 3 , . . .,t′ n ),where the complex algebraic function G ′ 2,2 (t′ ; ǫ 2 ) satisfies∂ ǫ2 G ′ 2,2(0; ǫ 2 )| ǫ2 =0 = 1 and G ′ 2,2(t ′ ; 0) ≡ t ′ 2.We also have to consider the modification of the first family of biholomorphismsG ′ 1 (t′ ; ǫ 1 ) := Φ 2 (G 1 (Φ ′ 2 (t′ ); ǫ 1 )). Using in an essential way thecommutativity, we may compute⎧Φ ′ 2(G ′ 1(t ′ ; ǫ 1 )) = G 1 (Φ ′ 2(t ′ ); ǫ 1 )(3.12)⎪⎨⎪⎩It follows that= G 1 (G 2 (t ′ 1 , 0, t′ 3 , . . .,t′ n ; t′ 2 ); ǫ 1)= G 2 (G 1 (t ′ 1, 0, t ′ 3, . . .,t ′ n; ǫ 1 ); t ′ 2)= G 2 (G 1,1 (t ′ 1 , 0, t′ 3 , . . .,t′ n ; ǫ 1), 0, t ′ 3 , . . .,t′ n ; t′ 2 )= Φ ′ 2 (G 1,1(t ′ 1 , 0, t′ 3 , . . .,t′ n ; ǫ 1), t ′ 2 , t′ 3 , . . ., t′ n ).(3.13) G ′ 1 (t′ ; ǫ 1 ) ≡ (G 1,1 (t ′ 1 , 0, t′ 3 , . . .,t′ n ; ǫ 1), t ′ 2 , t′ 3 , . . .,t′ n )whence(3.14) G ′ 1,1 (t′ ; ǫ 1 ) := G 1,1 (t ′ 1 , 0, t′ 3 , . . ., t′ n ; ǫ 1)does not depend on t ′ 2. Finally, inserting (3.11) and (3.13) in the commutativityrelation G ′ 1 (G′ 2 (t′ ; ǫ 2 ); ǫ 1 ) ≡ G ′ 2 (G′ 1 (t′ ; ǫ 1 ); ǫ 2 ), we find{G′1,1 (t ′ 1 , G′ 2,2 (t′ ; ǫ 2 ), t ′ 3 , . . .,t′ n ; ǫ 1) ≡ G ′ 1,1 (t′ ; ǫ 1 ),(3.15)G ′ 2,2(G ′ 1,1(t ′ ; ǫ 1 ), t ′ 2, t ′ 3, . . .,t ′ n; ǫ 2 ) ≡ G ′ 2,2(t ′ ; ǫ 2 ).

The first relation gives nothing, since we already know that G ′ 1,1 is independentof t ′ 2. By differentiating the second relation with respect to ǫ 1 at ǫ 1 = 0,we find that G ′ 2,2 is independent of t ′ 1.In summary, after the change of coordinates Φ ′ 2 ◦ Φ′ 1 (t′ ) = t which istangent to the identity map at t ′ = 0, we obtained that107(3.16){G′1 (t ′ ; ǫ 1 ) = (G ′ 1,1(t ′ 1, t ′ 3, . . .,t ′ n), t ′ 2, t ′ 3, . . .,t ′ n),G ′ 2 (t′ ; ǫ 2 ) = (t ′ 1 , G′ 2,2 (t′ 2 , t′ 3 , . . .,t′ n ), t′ 3 , . . .,t′ n ).Using these arguments, the proof of Proposition 3.1 clearly follows by induction.Now, we come back to our CR manifold M ′ having the one-parameterfamilies of algebraic biholomorphisms G ′ i(t ′ ; e i ) given by (3.2) and pairwisecommuting. Applying Proposition 3.1, after a change of complex algebraiccoordinates of the form t ′ = Ψ ′′ (t ′′ ), we may assume that the G ′′i (t′′ ; ǫ i ) arealgebraic and can be written in the specific form(3.17) G ′′i (t′′ ; ǫ i ) ≡ (t ′′1 , . . .,t′′ i−1 , G′′ i,i (t′′ i ; ǫ i), t ′′i+1 , . . .,t′′ n ),with ∂ ǫi G ′′i,i(0; ǫ i )| ǫi =0 = 1. Let t ′′ = Ψ ′ (t ′ ) denote the inverse of t ′ =Ψ ′′ (t ′′ ). We thus have t ′′ = Ψ ′ (t ′ ) = Ψ ′ (Φ(t)), where we remind that t ′ =Φ(t) provides the equivalence between the strong tube M and the algebraicCR generic M ′ .Since Ψ ′ is algebraic, the image M ′′ := Ψ ′ (M ′ ) is also algebraic. Letr j ′(t′ , ¯t ′ ) = 0, j = 1, . . .,d, be defining equations for M ′ . Then r j ′′(t′′, ¯t ′′ ) :=r j(Ψ ′ ′′ (t ′′ ), Ψ ′′ (t ′′ )) = 0 are defining equations for M ′′ . By assumption, forǫ i := e i ∈ R real, the family of algebraic biholomorphisms G ′ i (t′ ; ǫ i ) mapsa small piece of M ′ through the origin into M ′ . It follows trivially thatG ′′i (t ′′ ; ǫ i ) ≡ Ψ ′ (G ′ i(Ψ ′′ (t ′′ ); ǫ i )) maps a small piece of M ′′ through the origininto M ′′ . Furthermore, since dΨ ′′ (0) = Id, it follows that if we denoteX i ′′ := Ψ ′ ∗ (X′ i ), then X′′ i | 0 = ∂ t ′′i| 0 .Next, thanks to the specific form (3.17), by differentiating∂ ǫi G ′′i (t ′′ ; ǫ i )| ǫi =0, we get n vector fields of the form Z i ′′ = c ′′i (t ′′i ) ∂ t ′′By construction, the functions c ′′i (t ′′i ) are algebraic and satisfy c ′′i (0) = 1.Differentiating with respect to e i the identity r j ′′ (G ′′i (t ′′ ; e i ), G ′′i (t′′ ; e i )) = 0for r j ′′(t′′, ¯t ′′ ) = 0, i.e. for t ′′ ∈ M ′′ , we see that Z i ′′ is tangent to M ′′ ,i.e. we see that Z i ′′ is an infinitesimal CR automorphism of M ′′ . Consequently,there exist real constants λ i,l such that Z i ′′ = ∑ nl=1 λ i,l X l ′′.SinceZ i ′′ | 0 = X i ′′ | 0 = ∂ t ′′i| 0 , we have in fact λ i,l = 1 for i = l and λ i,l = 0 fori ≠ l. So Z i ′′ = X i ′′ and we have shown that(3.18) (Ψ ′ ◦ Φ) ∗ (X i ) = X ′′i = Z ′′i = c ′′i (t′′ i ) ∂ t ′′i .i.

108We shall call a CR generic manifold M ′′ having infinitesimal CR automorphismsof the form X i ′′ = c ′′i (t′′ i ) ∂ t ′′ with c ′′i i (0) ≠ 0 a pseudotube. Sucha pseudotube is not in general a product by R n . In fact, there is no hopeto tubify all algebraic peudotubes in algebraic coordinates, as shows the elementaryexample Im w = |z + 1| 2 + |z + 1| 6 − 2 having infinitesimalCR automorphisms ∂ w and i(z + 1)∂ z , since the only change of coordinatesfor which Φ ∗ (∂ w ) = ∂ w ′ and Φ ∗ (i(z + 1)∂ z ) = ∂ z ′ is z + 1 = e iz′ ,w = w ′ , which transforms M into M ′ of nonalgebraic defining equationIm w ′ = e −2y′ + e −6y′ .The constructions of this paragraph may be represented by the followingsymbolic picture.C n t v x, uMyΦC n v t ′′M ′x ′ , u ′y ′M strong tubeM ′ algebraicΨ ′C n t ′′ v ′′ x ′′ , u ′′′Φ ′′ := Ψ ′ ◦ ΦM ′′y ′′M ′′ algebraic pseudotubeSummary and conclusion of the first step. To conclude, let us denote forsimplicity M ′′ again by M ′ , the coordinates t ′′ again by t ′ and t ′′ = Ψ ′ ◦Φ(t) by t ′ = Φ(t). After the above straightenings, we have shown thatthe infinitesimal CR automorphisms X i ′ := Φ ∗(X i ) of the algebraic genericmanifold M ′ are of the sympathetic form X i ′ = c′ i (t′ i ) ∂ t ′ , i = 1, . . ., n, withialgebraic coefficients c ′ i(t ′ i) satisfying c ′ i(0) = 1.3.2. Proof of the second step. We characterize first finite nondegeneracyfor tubes of codimension d in C n .Lemma 3.2. Let M be a tube of codimension d in C n equipped with coordinates(z, w) = (x + iy, u + iv) ∈ C m × C d given by the equationsv j = ϕ j (y), j = 1, . . ., d, where ϕ j (0) = 0. Then M is finitely nondegenerateat the origin if and only if there exist m multi-indices β∗ 1, . . .,βm ∗ ∈ Nm

with |β∗ k | ≥ 1 and integers j∗, 1 . . .,j∗ m with 1 ≤ j∗ k ≤ d such that the realmapping( )∂|β∗ 1| ϕ j 1(3.19) ψ(y) :=∗(y), . . ., ∂|βm ∗ | ϕ j m ∗(y)=: y ′ ∈ R m∂y β1 ∗ ∂y βm ∗is of rank m at the origin in R m .Proof. We follow the definition of finite nondegeneracy given in §1.2. Letr j (t, ¯t) := v j − ϕ j (y) = 0 be the defining equations of M. Let L k :=∂¯zk + ∑ dj=1 ϕ j,¯z k∂ ¯wj , k = 1, . . .,m, be a basis of (1, 0)-vector fields tangentto∑M. We write the first order terms in the Taylor series of ϕ j (y) as ϕ j (y) =nl=1 λ j,l y l + O(|y| 2 ). Then the holomorphic gradient of r j is given by(3.20) ⎧∇ ⎪⎨ t (r j ) = (∂ z1 r j , . . .,∂ zm r j , ∂ w1 r j , . . .,∂ wd r j )= i2 −1 (∂ y1 ϕ j , . . .,∂ ym ϕ j , 0, . . .,0, −1, 0, . . ., 0)⎪⎩= i2 −1 (λ j,1 , · · · , λ j,m , 0, . . ., 0, −1, 0, . . ., 0), at the origin.On the other hand, since for β = (β 1 , . . .,β m ) ∈ N m with |β| ≥ 1 theorder |β| derivation L β := L β 11 · · ·Lβm m acts on functions of y as the operator(2i) −|β| ∂y β , we can compute(3.21) {βL (∇t (r j )) = (L β ∂ z1 ϕ j , . . ., L β ∂ zm ϕ j , 0, . . .,0, . . ., 0)= i −|β|+1 2 −|β|−1 (∂ β y ∂ y 1ϕ j , . . .,∂ β y ∂ y mϕ j , 0, . . ., 0, . . .,0).By inspecting the expressions (3.20) and (3.21), we see thatSpan{(L β (∇ t (r j )))(0) : β ∈ N m , j = 1, . . .,d} = C n if and onlyif Span{(∂ β y ∂ y1 ϕ j (0), . . ., ∂ β y ∂ ym ϕ j (0)) : β ∈ N m , |β| ≥ 1, j =1, . . ., d} = R m . This last condition is clearly equivalent to the one statedin Lemma 3.2.We can prove now that the inverse mapping ψ ′ (y ′ ) of the mappingψ(y) defined by (1.1) (or (3.19)) has algebraic first order derivatives.By Step 1, there exists a biholomorphic transformation Φ mapping thestrong tube M onto the algebraic pseudotube M ′ with the property thatΦ ∗ (∂ ti ) = c ′ i(t ′ i) ∂ t ′∑ i. Writing Φ(t) = (h 1 (t), . . .,h n (t)), we have Φ ∗ (∂ ti ) =nl=1 h l,t i(t) ∂ t ′l= c ′ i (t′ i ) ∂ t ′ , so h ii(t) depends only on t i which yieldsΦ(t) = (h 1 (t 1 ), . . .,h n (t n )). We shall use the convenient notation t ′ i =h i (t i ) and t i = h ′ i(t ′ i) for the inverse h ′ i := h −1i , i = 1, . . ., n. If accordingly,Φ ′ (t ′ ) = t denotes the inverse of Φ(t) = t ′ , we have Φ ′ ∗ (c′ i (t′ i ) ∂ t ′) = ic ′ i (t′ i ) h′ i,t(t ′ ′ i ) ∂ t i= ∂ ti , which shows that c ′ i (t′ i ) h′ ii,t i(t ′ ′ i ) ≡ 1. Since c′ i (0) =1, we see that h ′ i (t′ i ) = ∫ t ′ i0 1/[c′ i (σ)] dσ is the complex primitive of an algebraicfunction. This observation will be important.109

110After a permutation of the coordinates, we may assume that M ′ is givenin the coordinates t ′ = (z ′ , w ′ ) ∈ C m × C d by the real defining equationsIm w j ′ = ϕ′ j (z′ , ¯z ′ , Re w ′ ), j = 1, . . ., d, where the functions ϕ ′ j are algebraicand vanish at the origin. Solving in terms of w ′ by means of the algebraicimplicit function theorem, we can represent M ′ by the algebraic complexdefining equations(3.22) w ′ j = Θ′ j (z′ , ¯z ′ , ¯w ′ ), j = 1, . . .,d,where Θ ′ satisfies the vectorial functional equation w ′ ≡Θ ′ (z ′ , ¯z ′ , Θ ′ (¯z ′ , z ′ , w ′ )) (which we shall not use). According to thesplitting (z ′ , w ′ ) of coordinates, it is convenient to modify our previous notationby writing z k = f k ′(z′ k ), k = 1, . . .,m and w j = g j ′(w′ j ), j = 1, . . ., dinstead of t i = h ′ i (t′ i ), i = 1, . . .,n, and also{X′k = a ′ k(3.23)(z′ k ) ∂ z k ′ , k = 1, . . .,m, a′ k (0) = 1,Y j ′ = b′ j (w′ j ) ∂ w j ′ , j = 1, . . .,d, b′ j (0) = 1,instead of X i ′ = c′ i (t′ i ) ∂ t ′ . The relation c′ i i (t′ i ) h′ i,t i(t ′ ′ i ) ≡ 1 rewrites down inthe form{ a′k (z k) ′ f ′ k,z k ′ (z′ k) ≡ 1,(3.24)b ′ j (w′ j ) g′ j,w ′ j (w′ j ) ≡ 1.We remind that the derivatives of the f k ′ and of the g′ j are algebraic. Let nowt ′ = (z ′ , w ′ ) ∈ M ′ , thus satisfying (3.22). Then h ′ (t ′ ) = (f ′ (z ′ ), g ′ (w ′ ))belongs to M, namely we have for j = 1, . . ., d:(3.25)g ′ j (w′ j ) − ḡ′ j ( ¯w′ j )2i( f′= ϕ 1 (z 1 ′ ) − ¯f 1 ′(¯z′ 1 )j , . . ., f m ′ (z′ m ) − ¯f m ′ (¯z′ m ) ),2i2iwhere i = √ −1 here. Replacing w ′ j by Θ′ j (z′ , ¯z ′ , ¯w ′ ) in the left hand side,we get the following identity between converging power series of the 2m+dcomplex variables (z ′ , ¯z ′ , ¯w ′ ):(3.26)g ′ j(Θ ′ j(z ′ , ¯z ′ , ¯w ′ )) − ḡ ′ j( ¯w ′ j)2i( f′≡ ϕ 1 (z 1) ′ − ¯f 1(¯z ′ 1)′j , . . ., f m(z ′ m) ′ − ¯f )m(¯z ′ m)′ .2i2iLet us differentiate this identity with respect to z k ′ , for k = 1, . . ., m. Takinginto account the relations (3.24), we obtain(3.27)a ′ k (z′ k ) Θ′ j,z ′ k (z′ , ¯z ′ , ¯w ′ )b ′ j (Θ′ j (z′ , ¯z ′ , ¯w ′ ))≡ ∂ϕ (j f′1 (z 1 ′) − ¯f 1 ′(¯z′ 1 )∂y k 2i, . . ., f m ′ (z′ m ) − ¯f m ′ (¯z′ m ) ).2i

Clearly, the left hand side is an algebraic function A j,k ′ (z′ , ¯z ′ , ¯w ′ ). Then differentiatingagain with respect to the variables z k ′ the relations (3.27), we seethat for every multi-index β ∈ N m with |β| ≥ 1, and every j = 1, . . ., d,there exists an algebraic function A j,β ′ (z′ , ¯z ′ , ¯w ′ ) such that the followingidentity holds:(3.28)A j,β ′ (z′ , ¯z ′ , ¯w ′ ) ≡ ∂β 1+···+β mϕ j∂y β 11 · · ·∂βm( f′1 (z 1) ′ − ¯f 1(¯z ′ 1)′y m2i111, . . ., f m(z ′ m) ′ − ¯f )m(¯z ′ m)′ .2iDifferentiating (3.28) with respect to ¯w ′ , we see immediately that A j,β ′ isin fact independent of ¯w ′ . Furthermore, we see that A j,β ′ is real, namelyA j,β ′ (z′ , ¯z ′ ) ≡ A ′ j,β(¯z ′ , z ′ ). Now we extract from (3.28) the m identitieswritten for β := β∗ k, j := jk ∗ , k = 1, . . .,m and we use the invertibility ofthe mapping ψ defined in (3.19) (recall that ψ ′ (y ′ ) = y denotes the inverseof y ′ = ψ(y)), which yields(3.29)f ′ k (z′ k ) − ¯f ′ k (¯z′ k )2i≡ ψ ′ k (A ′j 1 ∗,β 1 ∗ (z′ , ¯z ′ ), . . .,A ′j m ∗ ,β m ∗ (z′ , ¯z ′ )),for k = 1, . . .,m. For simplicity, we shall write A k ′(z′, ¯z ′ ) instead ofA ′j∗ ∗(z ′ , ¯z ′ ). Finally, we differentiate (3.29) with respect to z ′ k,βk k , whichyields, taking into account (3.24):(3.30) ⎧⎪⎨⎪⎩12i a ′ k (z′ k ) ≡0 ≡m∑l=1m∑l=1∂ψ k′ (A ′∂yl′1 (z′, ¯z ′ ), . . ., A m ′ (z′, ¯z ′ )) ∂A l′(z ′ , ¯z ′ ),∂z ′ k∂ψ k′ (A∂y1(z ′ ′ , ¯z ′ ), . . ., A m(z ′ ′ , ¯z ′ )) ∂A l′(z ′ , ¯z ′ ), ˜k ≠ k.l′ ∂ze ′ kIt follows from these relations (3.30) viewed in matrix form that the constantmatrix ( ∂A l′(0, 0))∂z k′ 1≤l,k≤m is invertible, because the diagonal matrix(δ e kk [2ia ′ k (z′ k )]−1 ) 1≤k, e k≤mis evidently invertible at z k ′ = 0 (recall a′ k (0) = 1).Consequently, there exist algebraic functions B k,l ′ (z′ , ¯z ′ ) so that(3.31)∂ψ k′ (A ′∂yl′ 1 (z′ , ¯z ′ ), . . ., A m ′ (z′ , ¯z ′ )) ≡ B k,l ′ (z′ , ¯z ′ ).Next, setting ỹ k ′ = A k ′(iy′, −iy ′ ), k = 1, . . ., m we see, from the invertibilityof the matrix ( ∂A k′ (0, 0))∂z l′ 1≤k,l≤m and from the reality of A k ′(z′, ¯z ′ ), that theJacobian determinant at the origin of the mapping y ′ ↦→ A ′ (iy ′ , −iy ′ ) = ỹ k′is nonzero. Thus there are real algebraic functions C k ′ so that we can express

112y ′ in terms of ỹ ′ as y ′ k = C ′ k (ỹ′ ). Finally, we get∂ψ k′ (3.32) (ỹ ′∂yl′ 1 , . . .,ỹ′ m ) = B′ k,l (iC ′ (ỹ ′ ), −iC ′ (ỹ ′ )),where the right hand sides are algebraic; this shows that the partial derivatives∂ y ′ lψ k ′ are algebraic functions of ỹ′ .To obtain the equivalent formulation of Theorem 1.1, we observe the following.Lemma 3.3. For every k, l = 1, . . .,m, the functions ∂ y ′kψ ′ l (y′ ) arealgebraic functions of y ′ if and only if for every k 1 , k 2 = 1, . . .,m,the second derivative ∂ 2 y k1 y k2(y) is an algebraic function of ψ(y) =(∂ y1 ϕ(y), . . ., ∂ ym ϕ(y)).Proof. Differentiating the identities y k ≡ ψ k ′ (ψ(y)), k = 1, . . ., m, withrespect to y l , we getm∑m∑(3.33) δk l ≡ ∂ y ′jψ k(ψ(y)) ′ ∂ yj ψ j (y) ≡ ∂ y ′jψ k(y ′ ′ ) ∂y 2 my lϕ(y).j=1Applying Cramer’s rule, we see that there exist universal rational functionsR k,l such that{ ∂2yk y lϕ(y) ≡ R k,l ({∂ y ′k2ψ k ′ 1(y ′ )} 1≤k1 ,k 2 ≤m})(3.34)j=1≡ R k,l ({∂ y ′k2ψ ′ k 1(∂ y1 ϕ(y), . . ., ∂ ym ϕ(y))} 1≤k1 ,k 2 ≤m}).This implies the equivalence of Lemma 3.3.In conclusion, taking Theorem 2.1 for granted, the proof of Theorem 1.1is now complete.3.3. Proof of Theorem 1.5. Let M : v = ϕ(z, ¯z) be a rigid Levi nondegeneratehypersurface in C n passing through the origin. We may assume thatv = ∑ n−1k=1 ε k |z k | 2 +ϕ 3 (z, ¯z), where ε k = ±1 and we may write ϕ 3 (z, ¯z) =∑ n−1k=1 [¯z k ϕ 3 k (z) + z k ¯ϕ 3 k (¯z)] + ϕ4 (z, ¯z), with ϕ 4 (0, ¯z) ≡ ϕ 4 z k(0, ¯z) ≡ 0 andϕ 3 k = O(2). After making the change of coordinates z′ k := z k + ε k ϕ 3 k (z),w ′ := w, we come to the simple equation v ′ = ∑ n−1k=1 ε k |z k ′ |2 + χ ′ (z ′ , ¯z ′ ),where χ ′ (0, ¯z ′ ) ≡ χ z ′k(0, ¯z ′ ) ≡ 0, considered in Theorem 1.5.Assume that M is strongly rigid, locally algebraizable and let M ′ be analgebraic equivalent of M. Let t ′ = h(t) be such an equivalence, or in ourprevious notation z ′ = f(z, w) and w ′ = g(z, w). We note z = f ′ (z ′ , w ′ )and w = g ′ (z ′ , w ′ ) the inverse equivalence. Since M is strongly rigid,namely Hol(M) is generated by the single vector field X 1 := ∂ w , it followsthat Hol(M ′ ) is also one-dimensional, generated by the single vector

field X 1 ′ := h ∗ (X 1 ). Taking again Theorem 2.1 for granted and proceedingas in the first step of the proof of Proposition 3.1, we may algebraicallystraighten the complex foliation induced by X 1 ′ to the “vertical” foliationby w ′ -lines. Equivalently, we may assume that X 1 ′ = b′ (z ′ , w ′ ) ∂ w ′ with b ′algebraic and b ′ (0) = 1. The assumption h ∗ (∂ w ) = b ′ (z ′ , w ′ ) ∂ w ′ yields thatf ′ (z ′ , w ′ ) is independant of w ′ and that b ′ (z ′ , w ′ ) g w ′ ′(z′ , w ′ ) ≡ 1, so that asin (3.24) above, the derivative gw ′ is algebraic. Let ′w′ = Θ ′ (z ′ , ¯z ′ , ¯w ′ ) bethe complex defining equation of M ′ in these coordinates. The assumptionh ′ (M ′ ) = M yields the following power series identity(3.35) g ′ (z ′ , Θ ′ (z ′ , ¯z ′ , ¯w ′ )) − ḡ ′ (¯z ′ , ¯w ′ ) ≡ 2i ϕ(f ′ (z ′ ), ¯f ′ (¯z ′ )).By differentiating this identity with respect to z k ′ , we get(3.36)∂ z ′kg ′ (z ′ , Θ ′ (z ′ , ¯z ′ , ¯w ′ ))+ ∂ z k ′ Θ′ (z ′ , ¯z ′ , ¯w ′ n−1)b ′ (Θ ′ (z ′ , ¯z ′ , ¯w ′ )) ≡ 2i ∑l=1113∂ zl ϕ(f ′ (z ′ ), ¯f ′ (¯z ′ )) ∂ z ′kf ′l(z ′ ).We notice that the second term in the left hand side of (3.36) is algebraic.By differentiating in turn (3.36) with respect to ¯z k ′ and using the algebraicityof ∂ 2 zg ′ (z ′ , ¯Θ ′ (z ′ , ¯z ′ , ¯w ′ )), we obtain that there exist algebraic functionsk ′ w′A k ′1 ,k 2(z ′ , ¯z ′ ) such that(3.37)A ′k 1 ,k 2(z ′ , ¯z ′ ) ≡∑n−1l 1 ,l 2 =1∂ 2 z l1 ¯z l2ϕ(f ′ (z ′ ), ¯f ′ (¯z ′ )) ∂ z ′k1f ′l 1(z ′ ) ∂¯z ′k2¯f′l2(¯z ′ ).Without loss of generality, we may assume that h ′ is tangent to the identitymap at t ′ = 0. Then setting ¯z ′ := 0 in (3.37) and using the fact that∂z 2 l1 ¯z l2ϕ(z, 0) = δ l 2l1ε l1 + ∂z 2 l1 ¯z l2χ(z, 0) ≡ δ l 2l1ε l1 by the properties of χ in′Theorem 1.5 we get, since ∂¯z ′k2¯f l2(0) = δ k 2l 2:(3.38) A ′k 1 ,k 2(z ′ , 0) ≡ ε k2 ∂ z ′k1f ′ k 2(z ′ ),which shows that all the first order derivatives ∂ z ′kf l ′(z′) are algebraic.Next, since the canonical transformation to normalizing coordinates is algebraicand preserves the “horizontal” coordinates z ′ (cf. [CM1974]), hencedoes not perturb the complex foliation induced by X 1, ′ we may also assumethat M ′ is given in normal coordinates, namely that the function Θ ′ satisfiesΘ ′ (0, ¯z ′ , ¯w ′ ) ≡ Θ ′ (z ′ , 0, ¯w ′ ) ≡ ¯w ′ . Since the coordinates are normal forboth M and M ′ , it follows by setting ¯z ′ := 0 and ¯w ′ := 0 in (3.35) thatg ′ (z ′ , 0) ≡ 0. Consequently, ∂ z ′kg ′ (z ′ , 0) ≡ 0. Finally, by setting z ′ := 0 and¯w ′ := 0 in (3.36), we see that the first term in the left hand side vanishes andthat the second term is algebraic with respect to ¯z ′ , so we obtain that there

114exist algebraic functions B ′ k (¯z′ ) such that(3.39) B ′ k (¯z′ ) ≡∑n−1l=1∂ zl ϕ(0, ¯f ′ (¯z ′ )) ∂ z ′kf ′l (0) ≡ ε k ¯f ′ k (¯z′ ).We have proved that the components f k ′(z′ ) are all algebraic.Finally, coming back to the relation (3.36), we want to prove that thederivatives ∂ zl ϕ(f ′ (z ′ ), ¯f ′ (¯z ′ )) are all algebraic. However, the first termof (3.36) is not algebraic in general. Fortunately, using the fact thatΘ ′= ¯w ′ + O(2), we see that there exists a unique algebraic solution¯w ′ = Λ ′ (z ′ , ¯z ′ ) of the implicit equation Θ ′ (z ′ , ¯z ′ , ¯w ′ ) = 0, namely satisfyingΘ ′ (z ′ , ¯z ′ , Λ ′ (z ′ , ¯z ′ )) ≡ 0. Then by replacing ¯w ′ by Λ ′ in (3.36), we getthat there exist algebraic functions C k ′(z′ , ¯z ′ ) such that(3.40) C ′ k(z ′ , ¯z ′ ) ≡∑n−1l=1∂ zl ϕ(f ′ (z ′ ), ¯f ′ (¯z ′ )) ∂ z ′kf ′l(z ′ ).Since f ′ is tangent to the identity map, we can solve by Cramer’srule this linear system for the derivatives ∂ zl ϕ, which yields that the∂ zl ϕ(f ′ (z ′ ), ¯f ′ (¯z ′ )) are all algebraic. Since f ′ (z ′ ) is also algebraic, weobtain in sum that the derivatives ∂ zl ϕ(z, ¯z) are all algebraic. In conclusion,taking Theorem 2.1 for granted, the proof of Theorem 1.5 is complete.3.4. Proof of Theorem 1.4. Let M : v = ϕ(z¯z) in C 2 with Hol(M) generatedby ∂ w and iz∂ z . Without loss of generality, we can assume thatϕ(r) = r + O(r 2 ). Let M ′ be an algebraic equivalent of M. Let t = h ′ (t ′ ),or z = f ′ (z ′ , w ′ ), w = g ′ (z ′ , w ′ ) be a local holomorphic equivalence satisfyingh ′ (M ′ ) = M. Let t ′ = h(t) be its inverse. Then Hol(M ′ ) istwo-dimensional and generated by h ∗ (∂ w ) and h ∗ (iz∂ z ). First of all, usingthe algebraicity of the CR automorphism group of M ′ and proceedingas in the proof of Proposition 3.1, we can prove that there exist two generatorsof Hol(M ′ ) of the form X 1 ′ = b ′ (w ′ ) ∂ w ′ and X 2 ′ = a ′ (z ′ ) ∂ z ′ whereb ′ and a ′ are algebraic and satisfy b ′ (0) = 1 and a ′ (z ′ ) = iz ′ + O(z ′2 ).Furthermore, we may assume that h ′ is tangent to the identity map and thath ′ ∗ (b′ (w ′ ) ∂ w ′) = ∂ w and h ′ ∗ (a′ (z ′ ) ∂ z ′) = iz∂ z . As in (3.24), it follows thatb ′ (w ′ ) g w ′ ′(w′ ) ≡ 1 and a ′ (z ′ ) f z ′ ′(z′ ) ≡ if ′ (z ′ ). Let w ′ = Θ ′ (z ′ , ¯z ′ , ¯w ′ ) be thecomplex algebraic equation of M ′ . Then we get the following power seriesidentity:(3.41) g ′ (Θ ′ (z ′ , ¯z ′ , ¯w ′ )) − ḡ ′ ( ¯w ′ ) ≡ 2i ϕ(f ′ (z ′ ) ¯f ′ (¯z ′ )),

which yields after differentiating with respect to z ′ :(3.42) {Θ′z ′(z′ , ¯z ′ , ¯w ′ )/[b ′ (Θ ′ (z ′ , ¯z ′ , ¯w ′ ))] ≡ 2i ∂ z ′f(z ′ ) ¯f ′ (¯z ′ ) ∂ r ϕ(f ′ (z ′ ) ¯f ′ (¯z ′ ))115≡ − 2 f ′ (z ′ ) ¯f ′ (¯z ′ ) ∂ r ϕ(f ′ (z ′ ) ¯f ′ (¯z ′ ))/[a ′ (z ′ )].Here, we consider the function ϕ as a function ϕ(r) of the real variabler ∈ R. Since the left hand side is an algebraic function and a ′ (z ′ ) is alsoalgebraic, there exists an algebraic function A ′ (z ′ , ¯z ′ ) such that we can write(3.43) A ′ (z ′ , ¯z ′ ) ≡ f ′ (z ′ ) ¯f ′ (¯z ′ ) ∂ r ϕ(f ′ (z ′ ) ¯f ′ (¯z ′ )).Next, using the property ϕ(r) = r + O(r 2 ), differentiating (3.43) withrespect to ¯z ′ at ¯z ′ = 0, we obtain that f ′ (z ′ ) is algebraic. Coming backto (3.43), this yields that ∂ r ϕ(f ′ (z ′ ) ¯f ′ (¯z ′ )) is algebraic. Since f ′ (z ′ ) is alsoalgebraic, we finally obtain that ∂ r ϕ(r) is algebraic. Excepting the exampleswhich will be treated in §7.5, the proof of Theorem 1.4 is complete.The next three sections are devoted to the statement of Theorem 4.1,which implies directly Theorem 2.1 (§4), and to its proof (§§5-6).§4. LOCAL LIE GROUP STRUCTURE FOR THE CR AUTOMORPHISMGROUP4.1. Local representation of a real algebraic generic submanifold. Weconsider a connected real algebraic (or more generally, real analytic) genericsubmanifold M in C n of codimension d ≥ 1 and CR dimension m =n − d ≥ 1. Pick a point p ∈ M and consider some holomorphic coordinatest = (t 1 , . . .,t n ) = (z 1 , . . .,z m , w 1 , . . .,w d ) ∈ C m × C d vanishingat p in which T 0 M = {Im w = 0}. If we denote w = u + iv, it followsthat there exists (Nash) real algebraic power series ϕ j (z, ¯z, u) withϕ j (0) = 0 and dϕ j (0) = 0 such that the defining equations of M are ofthe form v j = ϕ j (z, ¯z, u), j = 1, . . .,d in a neighborhood of the origin.By means of the algebraic implicit function theorem, we can solve with respectto ¯w the equations w j − ¯w j = 2i ϕ j (z, ¯z, (w + ¯w)/2), j = 1, . . ., d,which yields ¯w j = Θ j (¯z, z, w) for some power series Θ j which are complexalgebraic with respect to their 2m + d variables. Here, we haveΘ j = w j + O(2), since T 0 M = {Im w = 0}. Without loss of generality,we shall assume that the coordinates are normal, namely the functionsΘ j (¯z, z, w) satisfy Θ j (0, z, w) ≡ w j and Θ j (¯z, 0, w) ≡ w j . It may beshown that the power series Θ j = w j + O(2) satisfy the vectorial functionalequation Θ(¯z, z, Θ(z, ¯z, ¯w)) ≡ ¯w in C{z, ¯z, ¯w} d and conversely that to everysuch power series mapping satisfying this vectorial functional equation,there corresponds a unique real algebraic generic manifold M (cf. for instancethe manuscript [GM2001c] for the details). So we can equivalently

116take ¯w j = Θ j (¯z, z, w) or w j = Θ j (z, ¯z, ¯w) as complex defining equationsfor M.For arbitrary ρ > 0, we shall often consider the open polydisc ∆ n (ρ) :={t ∈ C n : |t| < ρ} where we denote by |t| := max 1≤i≤n |t i | the usualpolydisc norm. Without loss of generality, we may assume that the powerseries Θ j converge normally in the polydic ∆ 2m+d (2ρ 1 ), where ρ 1 > 0. Infact, we shall successively introduce some other positive constants (radii)0 < ρ 5 < ρ 4 < ρ 3 < ρ 2 < ρ 1 afterwards. Finally, we define M as:(4.1) M = {(z, w) ∈ ∆ n (ρ 1 ) : ¯w j = Θ j (¯z, z, w), j = 1, . . .,d}.Next, let ρ 2 arbitrary with 0 < ρ 2 < ρ 1 . For h ′ , h ∈ O(∆ n (ρ 1 ), C n ), wedefine(4.2) ||h ′ − h|| ρ2 := sup {|h ′ (t) − h(t)| : t ∈ ∆ n (ρ 2 )}.For k ∈ N, we shall also consider the C k norms(4.3)||J k h ′ −J k h|| ρ2 := sup {|∂ α t h′ (t)−∂ α t h(t)| : t ∈ ∆ n(ρ 2 ), α ∈ N n , |α| ≤ k}.For k ∈ N and t ∈ ∆ n (ρ 1 ), we denote by J k h(t) the collection of partialderivatives (∂t α h i (t)) 1≤i≤n, |α|≤k of length ≤ k of the components h 1 , . . .,h n ,so J k h(t) ∈ C N n,k, where N n,k := n (n+k)! . In particular, the expressionn! k!J k h(0) = (∂t αh i(0)) 1≤i≤n, |α|≤k denotes the k-jet of h at 0. So, the spaceof k-jets at the origin of holomorphic mappings h ∈ O(∆ n (ρ 1 ), C n ) maybe identified with the complex linear space C N n,k . We denote the naturalcoordinates on C N n,k by (Jαi ) 1≤i≤n, |α|≤k . Sometimes, we abbreviate thiscollection of coordinates by J k ≡ (Ji α ) 1≤i≤n, |α|≤k . Finally, we denote by JIdkthe k-jet at the origin of the identity mapping. We introduce the importantset of holomorphic self-mappings of M defined by(4.4) {Hρ 2 ,ρ 1M,k,ε := {h ∈ O(∆ n(ρ 1 ), C n ) : ||J k h − JId|| k ρ2 < ε,h(M ∩ ∆ n (ρ 2 )) ⊂ M ∩ ∆ n (ρ 1 )}.Here, k ∈ N and ε > 0 is a small positive number that we shall shrink manytimes in the sequel.

117Im w∆ n(ρ 4 )∆ n(ρ 1 )∆ n(ρ 2)∆ n(ρ 3)finitenondegeneracyassumptionMMRew, z, ¯z∆n(ρ 5 )FIGURE 3: NEST OF POLYDISCS CENTERED AT 0 ∈ MMinimalityassumptionρ 4 ∼ (ρ 1) NN >> 1We may now state the main theorem of §4, §5 and §6, namely Theorem4.1, which provides a complete parametrized description of the setH ρ 2,ρ 1M,k,εof local biholomorphic self-mappings of M, with k equal to an integerκ 0 depending on M. During the course of the (rather long) proof, fortechnical reasons, we shall have to introduce first a third positive radius ρ 3with 0 < ρ 3 < ρ 2 < ρ 1 which is related to the finite nondegeneracy of M,and then afterwards a fourth positive radius ρ 4 with 0 < ρ 4 < ρ 3 < ρ 2 < ρ 1 ,which is related to the minimality of M. This is why the radius notation“ρ 4 ” appears after “ρ 2 ” and “ρ 1 ” without mention of “ρ 3 ” (cf. FIGURE 3).Theorem 4.1. Assume that the real algebraic generic submanifold M definedby (4.1) is minimal and finitely nondegenerate at the origin. As above,fix two radii ρ 1 and ρ 2 with 0 < ρ 2 < ρ 1 . Then there exists an even integerκ 0 ∈ N ∗ which depends only on the local geometry of M near the origin,there exists ε > 0, there exists ρ 4 > 0 with ρ 4 < ρ 2 , there exists a complexalgebraic C n -valued mapping H(t, J κ 0) which is defined for t ∈ C n with|t| < ρ 4 and for J κ 0∈ C Nn,κ 0 (where Nn,κ0 = n (n+κ 0)!n! κ 0) with |J κ 0−J κ 0! Id | < εand which depends only on M and there exists a geometrically smooth realalgebraic totally real submanifold E of C Nn,κ 0 passing through the identityjet J κ 0Idwhich depends only on M, which is defined by(4.5) E = {J κ 0: |J κ 0− J κ 0Id | < ε, C l(J κ 0, J κ 0 ) = 0, l = 1, . . ., υ},where the C l (J κ 0, J κ 0 ), l = 1, . . ., υ, are real algebraic functions defined onthe polydisc {|J κ 0− J κ 0Id| < ε}, and which can be constructed algorithmicallyby means only of the defining equations of M, such that the followingsix statements hold:

118(1) Every local biholomorphic self-mapping h ∈ H ρ 2,ρ 1M,κ 0 ,εof M (which isdefined on the large polydisc ∆ n (ρ 1 )) is represented by(4.6) h(t) = H(t, J κ 0h(0)),on the smallest polydisc ∆ n (ρ 4 ). In particular, each h ∈ H ρ 2,ρ 1M,κ 0 ,ε isa complex algebraic biholomorphic mapping. Furthermore, the κ 0 -jetof h at the origin belongs to the real algebraic submanifold E, namelywe have C l (J κ 0h(0), J κ 0¯h(0)) = 0, l = 1, . . ., υ.(2) Conversely, shrinking ε if necessary, given an arbitrary jet J κ 0in Ethere exists a smaller positive radius ρ 5 < ρ 4 such that the mappingdefined by h(t) := H(t, J κ 0) for |t| < ρ 5 sends M ∩ ∆ n (ρ 5 ) CRdiffeomorphicallyonto its image which is contained in M ∩ ∆ n (ρ 4 ).We may therefore say that the set H ρ 2,ρ 1M,κ 0 ,εof local biholomorphic selfmappingsof M is parametrized by the real algebraic submanifold E.(3) For every choice of two smaller positive radii ˜ρ 1 ≤ ρ 1 and ˜ρ 2 ≤ ρ 2 with˜ρ 2 < ˜ρ 1 , there exists a positive radius ˜ρ 4 ≤ ρ 4 with ˜ρ 4 < ˜ρ 2 , and a positive˜ε ≤ ε such that the same complex algebraic mapping H(t, J κ 0)as in statement (1) above represents all local biholomorphic selfmappings˜h ∈ H eρ 2,eρ 1M,κ 0 ,eε of M, namely we have ˜h(t) = H(t, J κ 0˜h(0))for all |t| < ˜ρ 4 as in (4.6). Furthermore, the corresponding real algebraictotally real submanifold Ẽ coincides with E in the polydisc{|J κ 0− J κ 0Id| < ˜ε} and it is defined by the same real algebraic equationsC l (J κ 0, J κ 0 ) = 0, l = 1, . . .,υ, as in equation (4.5). In fact,the algebraic mapping H(t, J κ 0) and the real algebraic totally realsubmanifold E depend only on the local geometry of M in a neighborhoodof the origin, namely on the germ of M at 0.(4) The set H ρ 2,ρ 1M,κ 0 ,ε, equipped with the law of composition of holomorphicmappings, is a real algebraic local Lie group. More precisely, let thepositive integer c 0 denote the real dimension of E, which is independentof ρ 1 , ρ 2 and consider a parametrization(4.7) R c 0∋ e = (e 1 , . . .,e c0 ) ↦→ j κ0 (e) ∈ E ⊂ C Nn,κ 0of the real algebraic totally real submanifold E. Then there exist a realalgebraic associative local multiplication mapping (e, e ′ ) ↦→ µ(e, e ′ )and a real algebraic local inversion mapping e ↦→ ι(e) such that if wedefine H(t; e) := H(t, j κ0 (e)), then H(H(t; e); e ′ ) ≡ H(t; µ(e, e ′ ))and H(t; e) −1 ≡ H(t; ι(e)), with the local Lie transformation groupaxioms, as defined in §2.3, being satisfied by H, µ and ι.(5) For i = 1, . . .,c 0 , consider the one-parameter families of transformationsdefined by H(t; 0, . . ., 0, e i , 0, . . .,0) =: H i (t; e i ) =:H i,ei (t). Then for each i = 1, . . .,c 0 , the vector field X i | (t;ei ) :=

[∂ i,ei H ei (t ′ )] t ′ =He −1i(t) , is defined for t ∈ ∆ n(ρ 5 ) and |e i | < ε, has algebraiccoefficients depending on the “time” parameter e i , and hasan algebraic flow, since this coincides with the algebraic mapping(t, e i ) ↦→ H i,ei (t).(6) Let ρ 5 be as in statement (2). Then the dimension c 0 of the real Lie algebraHol(M, ∆ n (ρ 5 )) is finite, bounded by the fixed integer N n,κ0 :=n (n+κ 0)!n! κ 0. Furthermore, each vector field X ∈ Hol(M, ∆! n (ρ 5 )) hascomplex algebraic coefficients.If M is real analytic, the same theorem holds with the word “algebraic”replaced everywhere by the word “analytic”.We shall explain below how the integer κ 0 is related to the minimalityand to the finite nondegeneracy of M at the origin. The next §5 and §6are devoted to the proof Theorem 4.1, namely the existence of the mappingH(t, J κ 0), the existence of the real algebraic totally real submanifold E andthe completion of the proof of properties (1-6).119§5. MINIMALITY AND FINITE NONDEGENERACY5.1. Local CR geometry of complexified real analytic generic submanifolds.Let ζ ∈ C m and ξ ∈ C d denote some independent coordinates correspongingto the complexification of the variables ¯z and ¯w, which we denotesymbolically by ζ := (¯z) c and ξ := ( ¯w) c , where the letter “c” stands for theword “complexified”. We also write τ := (¯t) c , so τ = (ζ, ξ) ∈ C n . Theextrinsic complexification M := (M) c of M is the complex submanifold ofcodimension d defined by(5.1) M := {(z, w, ζ, ξ) ∈ ∆ n (ρ 1 ) × ∆ n (ρ 1 ) : ξ = Θ(ζ, z, w)}.If M is (real, Nash) algebraic, so is M . As remarked, we can choose theequivalent defining equation w = Θ(z, ζ, ξ) for M . In the remainder of §5,we shall essentially deal with M instead of M. In fact, M clearly imbedsin M as the intersection of M with the antiholomorphic diagonal Λ :={(t, τ) ∈ C n × C n : τ = ¯t}.Following [Me1998], [Me2001], we shall complexify a conjugatepair of generating families of CR vector fields tangent to M, namelyL 1 , . . ., L m of type (1, 0) and their conjugates L 1 , . . .,L m whichare of type (0, 1). Here, we can explicitely choose the generatorsL k = ∂/∂z k + ∑ dj=1 [∂Θ j/∂z k (z, ¯z, ¯w)] ∂/∂w j for k = 1, . . .,m. Thentheir complexification yields a pair of collections of m vector fields defined

120over ∆ n (ρ 1 ) × ∆ n (ρ 1 ) by⎧L k := ∂ +⎪⎨ ∂z k(5.2)⎪⎩ L k := ∂ +∂ζ kd∑j=1d∑j=1∂Θ j∂z k(z, ζ, ξ)∂∂w j,∂Θ j∂ζ k(ζ, z, w) ∂∂ξ j,k = 1, . . .,m,k = 1, . . .,m.The reader may check directly that L k (w j − Θ j (z, ζ, ξ)) ≡ 0, whichshows that the vector fields L k are tangent to M . Similarly, L k (ξ j −Θ j (ζ, z, w)) ≡ 0, so the vector fields L k are also tangent to M . Furthemore,we may check the commutation relations [L k , L k ′] = 0 and[L k , L k ′] = 0 for all k, k ′ = 1, . . ., m. It follows from the Frobeniustheorem that the two m-dimensional distributions spanned by each of thesetwo collections of m vector fields has the integral manifold property. Thisis not surprising since the vector fields L k are the vector fields tangent tothe intersection of M with the sets {τ = τ p = ct.}, which are clearly m-dimensional complex integral manifolds. Following [Me1998], [Me2001],we denote these manifolds by S τp := {(t, τ p ) : w = Θ(z, ζ p , ξ p )}, whereτ p is a constant, and we call them complexified Segre varieties. Similarly,the integral manifolds of the vector fields L k are the conjugate complexifiedSegre varieties S tp:= {(t p , τ) : ξ = Θ(ζ, z p , w p )}, where t p is fixed.The union of the manifolds S τp induces a local complex algebraic foliationF of M by m-dimensional leaves. Similarly, there is a second foliation Fwhose leaves are the S tp.The following symbolic picture summarizes our constructions. However,we warn the reader that the codimension d ≥ 1 of the union of the twofoliations F and F in M is not visible in this two-dimensional figure.Mτ pL0F{t = t p}S tpS τpFLt pΛ{τ = τ p}tThe complexification of a real analyticCR-generic manifold M carries twocomplex foliations F L and F L directedby the complefixied CR-vector fields L and Lwhose leaves coincide with the complexifiedSegre varieties S τp and S tp .These leaves also coincide with the intersectionof M with the horizontal slices {τ = τ p}and with the vertical slices {t = t p}.FIGURE 4: GEOMETRY OF THE COMPLEXIFICATION MNow, we introduce the “multiple” flows of the two collections of conjugatevector fields (L k ) 1≤k≤m and (L k ) 1≤k≤m . For an arbitrary point

p = (w p , z p , ζ p , ξ p ) ∈ M and for an arbitrary complex “multitime” parameterz 1 = (z 1,1 , . . .,z 1,m ) ∈ C m , we define(5.3) {Lz1 (z p , w p , ζ p , ξ p ) := exp(z 1 L )(p) := exp(z 1,1 L 1 (· · ·(exp(z 1,m L m (p))) · · ·)) :=:= (z p + z 1 , Θ(z p + z 1 , ζ p , ξ p ), ζ p , ξ p ).With this formal definition, there exists a maximal connected open subset Ωof M × C m containing M × {0} such that L z1 (p) ∈ M for all (z 1 , p) ∈ Ω.Analogously, for (ζ 1 , p) running in a similar open subset Ω, we may alsodefine the map(5.4) L ζ1(z p , w p , ζ p , ξ p ) := (z p , w p , ζ p + ζ 1 , Θ(ζ p + ζ 1 , z p , w p )).We notice that the two maps given by (5.3) and (5.4) are holomorphic in theirvariables. Since M is real algebraic, they are moreover complex algebraic.5.2. Segre chains. Let us start from the point p being the origin and let usmove alternately in the direction of S or of S , namely we consider the twomaps Γ 1 (z 1 ) := L z1 (0) and Γ 1 (z 1 ) := L z1(0). Next, we start from theseendpoints and we move in the other direction, namely, we consider the twomaps(5.5) Γ 2 (z 1 , z 2 ) := L z2(L z1 (0)), Γ 2 (z 1 , z 2 ) := L z2 (L z1(0)),where z 1 , z 2 ∈ C m . Also, we define Γ 3 (z 1 , z 2 , z 3 ) := L z3 (L z2(L z1 (0))),etc. By induction, for every positive integer k, we obtain two mapsΓ k (z 1 , . . .,z k ) and Γ k (z 1 , . . ., z k ). In the sequel, we shall often use the notationz (k) := (z 1 , . . .,z k ) ∈ C mk . Since Γ k (0) = Γ k (0) = 0, for everyk ∈ N ∗ , there exists a sufficiently small open polydisc ∆ mk (δ k ) centered atthe origin in C mk with δ k > 0 such that Γ k (z (k) ) and Γ k (z (k) ) belong to Mfor all z (k) ∈ ∆ mk (δ k ).We also exhibit a simple link between the maps Γ k and Γ k . Let σ bethe antiholomorphic involution defined by σ(t, τ) := (¯τ, ¯t). Since w =Θ(z, ζ, ξ) if and only if ξ = Θ(ζ, z, w), this involution maps M onto Mand it also fixes the antidiagonal Λ pointwise. Using the definitions (5.3)and (5.4), we see readily that σ(L z1 (0)) = L ¯z1 (0). It follows generally thatσ(Γ k (z (k) )) = Γ k (z (k) ).Next, we observe that Γ k+1 (z (k) , 0) = Γ k (z (k) ), since L 0 and L 0 coincidewith the identity map. So the ranks at the origin of the maps Γ k increase withk.Definition 5.1. The real analytic generic manifold M is said to be minimalat p if the maps Γ k are of (maximal possible) rank equal to 2m + d =dim C M at the origin in ∆ mk (δ k ) for all k large enough.The following fundamental properties are established in [Me1998],[Me2001].121

122Theorem 5.2. The minimality of M at 0 is a biholomorphically invariantproperty. It depends neither on the choice of a defining equation for Mnor on the choice of a system of generating complexified CR vector fields(L k ) 1≤k≤m and (L k ) 1≤k≤m . Also, minimality is equivalent to the fact thatthe Lie algebra generated by the complexified CR vector fields (L k ) 1≤k≤mand (L k ) 1≤k≤m spans TM in a neighborhood of 0. Furthermore, thereexists an invariant integer ν 0 , called the Segre type of M at 0 satisfyingν 0 ≤ d + 1 which is the smallest integer such that the mappings Γ k and Γ kare of generic rank equal to 2m+d over ∆ mk (δ k ) for all k ≥ ν 0 +1. Finally,with this integer ν 0 , the odd integer µ 0 := 2ν 0 + 1, called the Segre type Mat 0 is the smallest integer such that the mappings Γ k and Γ k are of rankequal to 2m + d at the origin in ∆ mk (δ k ).Let µ 0 := 2ν 0 + 1 be the Segre type of M at 0 (notice that this is alwaysodd). In the remainder of this section, we assume that M is minimal at 0 andwe exploit the rank condition on Γ k . More precisely we choose a positive ηwith 0 < η ≤ δ µ0 such that Γ µ0 has rank 2m+d at every point of the polydisc∆ mµ0 (η). Without loss of generality, we can also assume that Γ µ0 (∆ mµ0 (η))contains M ∩ (∆ n (ρ 4 ) × ∆ n (ρ 4 )). Simple examples in the hypersurfacecase show that ρ 4

Finite nondegeneracy is interesting for the following reason. In the sequel,we shall have to consider an infinite collection of equations of the form(5.7) Θ j,β (t) + ∑γ∈N m ∗γ (β + γ)!(ζ) Θ j,β+γ (t) = ω j,β ,β! γ!where N m ∗ := N m \{0}, where j runs from 1 to d, where β runs in N mand where the right hand sides ω j,β are independent complex variables. Forβ = 0, the equations (5.7) write simply Θ j (ζ, t) = ω j,0 . By definition,if M is l 0 -nondegenerate at 0, there exists n integers j∗, 1 . . .,j∗n with 1 ≤j∗ i ≤ d and n multi-indices β1 ∗ , . . ., βn ∗ ∈ N m with |β∗ i| ≤ l 0 such thatthe local holomorphic self-mapping t ↦→ (Θ j k ∗ ,β (t)) ∗ k 1≤k≤n of C n is of rankn at the origin. Considering the equations (5.7) for j = j∗ 1, . . .,jn ∗ andβ = β∗, 1 . . ., β∗ n and applying the implicit function theorem, we observe thatwe can solve t in terms of (ζ, ω j 1 ∗ ,β∗ 1, . . ., ω j∗ n ,β∗ n ) by means of a holomorphicmapping, namely(5.8) t = Ψ(τ, ω j 1 ∗ ,β 1 ∗ , . . .,ω j n ∗ ,βn ∗ ).Without loss of generality, we may assume that Ψ is holomorphic for |ζ|

124now differentiate (6.2) by applying the vector fields L 1 , . . .,L m , whichgivesm∑ ∂Θ j(6.3) L k ḡ j (τ) = (∂ζ ¯f(τ), h(t)) L k ¯fl (τ),ll=1for k = 1, . . ., m and j = 1, . . .,d. For fixed j, we consider the m equations(6.3) as an affine system satisfied by the partial derivatives ∂Θ j /∂ζ l .By Cramer’s rule, there exists universal polynomials Ω j,k in their variablessuch that(6.4)∂Θ j∂ζ k( ¯f(τ), h(t)) = Ω j,k({L l ¯h(τ)}1≤l≤m )det (L k ¯fl (τ)) 1≤k,l≤nfor all (t, τ) ∈ M with |t|, |τ| < ρ 2 and for k = 1, . . ., m, j = 1, . . .,d.Applying the derivations L k to (6.4) we see by induction that for everymulti-index β ∈ N m ∗ and for every j = 1, . . .,d, there exists a universalpolynomial Ω j,β in its variables such that(6.5)1 ∂ |β| Θ jβ! ∂ζ ( ¯f(τ), h(t)) = Ω j,β({L γ ¯h(τ)}|γ|≤|β| )β [det (L k ¯fl (τ)) 1≤k,l≤n ] 2|β|−1,for all (t, τ) ∈ M with |t|, |τ| < ρ 2 . Here, for γ = (γ 1 , . . .,γ m ) ∈ N m ,we denote by L γ the derivation (L 1 ) γ 1. . .(L m ) γm . Next, denoting byω j,β (t, τ) the right hand side of (6.5) and developing the left hand side inpower series using (5.7), we may write(6.6) Θ j,β (h(t)) + ∑( ¯f(τ)) γ Θ j,β+γ (h(t)) = ω j,β (t, τ).(6.7)h(t) = Ψ⎪⎨⎪⎩γ∈N m ∗Recall that M is l 0 -nondegenerate at 0. Using (5.8), we can solve h(t) interms of the derivatives of ¯h(τ), namely⎧ (¯f(τ),Ω j 1 ∗ ,β 1 ∗ ({L γ ¯h(τ)}|γ|≤|β 1 ∗ |)[det (L k ¯fl (τ)) 1≤k,l≤n ] 2|β1 ∗ |−1, . . .. . .,Ω j n ∗ ,β n ∗ ({L γ ¯h(τ)}|γ|≤|β n ∗ |)[det (L k ¯fl (τ)) 1≤k,l≤n ] 2|βn ∗ |−1= Ψ( ¯f(τ), ω j 1 ∗ ,β∗ 1(t, τ), . . .,ω j∗ n (t, τ)). ,βn ∗)=Here, the maximal length of the multi-indices β 1 ∗ , . . .,βn ∗ is equal to l 0. Accordingto (5.8), the representation (6.7) of h(t) holds provided |ḡ(τ)| < ˜ρ 3and |ω j i ∗ ,β i ∗ | < ˜ρ 3. Since the coordinates are normal, we have Θ j (¯z, 0, 0) ≡0, or equivalently Θ j,β (0) = 0 for all j = 1, . . .,d and all β ∈ N m . It followsfrom (6.6) and from h(0) = 0 that ω j,β (0) = 0, for all j = 1, . . .,d and allβ ∈ N m . Consequently, there exists a radius ρ 3 ∼ ˜ρ 3 with 0 < ρ 3 < ρ 2 < ρ 1

such that |ω j i ∗ ,β∗ i(t, τ)| < ˜ρ 3, i = 1, . . .,n and such that |ḡ(τ)| < ˜ρ 3 for allh ∈ H ρ 2,ρ 1M,κ 0 ,ε and for all (t, τ) ∈ M with |t|, |τ| < ρ 3.In conclusion, the relation (6.7) holds for all h ∈ H ρ 2,ρ 1M,κ 0 ,εand for all(t, τ) ∈ M with |t|, |τ| < ρ 3 .Next, using the explicit expressions of the vector fields L k given in (5.2),we may develop the higher order derivatives L γ¯h(τ) as polynomials in the|γ|-jet (∂τ γ′ ¯h(τ)) |γ ′ |≤|γ| of ¯h(τ) with coefficients being certain holomorphicfunctions of (t, τ) obtained as certain polynomials with respect to the partialderivatives of the functions Θ j (ζ, t).To be more explicit in this desired new representation of (6.7), we remindfirst our jet notation. For each i = 1, . . .,n and each α ∈ N n , we introduceda new independent coordinate Jiα corresponding to the partial derivative∂τ α¯h i (τ) (or ∂t αh i(t)). The space of k-jets of holomorphic mappings ¯h(τ) isthen the complex space C n(n+k)! n! k! with coordinates (Ji α) 1≤i≤n, |α|≤k. It will beconvenient to use the abbreviations J k := (Ji α) 1≤i≤n, |α|≤k and J k¯h(τ) :=(∂τ α¯h i (τ)) 1≤i≤n, |α|≤k .So pursuing with (6.7), we argue that for every γ ∈ N m , there existsa polynomial in the jet J |γ|¯h(τ) with holomorphic cooeficients dependingonly on Θ such that(6.8) L γ ¯h(τ) ≡ Pγ (t, τ, J |γ|¯h(τ)).Putting all these expressions in (6.7), we obtain an important relation betweenh and the l 0 -jet of ¯h which we may now summarize. At first, asκ 0 = l 0 (µ 0 + 1) ≥ l 0 , observe that for every h ∈ H ρ 2,ρ 1M,κ 0 ,ε, we have||J l 0h − J l 0Id || ρ 2≤ ||J κ 0h − J κ 0Id || ρ 2≤ ε. Shrinking ε if necessary, wehave proved the following lemma.Lemma 6.1. There exists a complex algebraic C n -valued mappingΠ(t, τ, J l 0) defined for |t|, |τ| < ρ 3 and for |J l 0− J l 0Id| < ε which dependsonly on the defining functions ξ j − Θ j (ζ, t) of M , such that for everylocal holomorphic self-mapping h ∈ H ρ 2,ρ 1M,κ 0 ,εof M (hence satisfying||J l 0h − J l 0Id || ρ 2< ε), the relation(6.9) h(t) = Π(t, τ, J l 0¯h(τ))holds for all (t, τ) ∈ M with |t|, |τ| < ρ 3 .6.2. Reflection identity for arbitrary jets. Let now Υ j and Υ j be the vectorfields tangent to M defined by(6.10)Υ j := ∂∂w j+d∑l=1Θ l,wj (ζ, t) ∂∂ξ l,Υ j := ∂∂ξ j+d∑l=1Θ l,ξj (z, τ) ∂∂w l,125

126for j = 1, . . ., d. We observe that the collection of 2m + d vector fieldsL k , L k , Υ j span TM . The same holds for the collection L k , L k , Υ j . Wealso have the commutation relations [Υ j , L k ] = 0 and [Υ j , L k ] = 0. Weobserve that Υ γ h(t) = ∂wh(t) γ for all γ ∈ N d . Let α = (β, γ) ∈ N m × N d .By expanding L β Υ γ h(t) using the explicit expressions (5.2), we obtaina polynomial Q β,γ (t, τ, (∂t α′ h(t)) |α ′ |≤|α|), where Q β,γ is a polynomial in itslast variables with coefficients depending on Θ and its partial derivatives.Conversely, since L k | 0 = ∂ zk at the origin, we can invert these formulas, sothere exist polynomials P α in their last variables with coefficients dependingonly on Θ such that(6.11) ∂ α t h(t) = P α (t, τ, (L β′ Υ γ′ h(t)) |β ′ |≤|β|, |γ ′ |≤|γ|).Lemma 6.2. For every l ∈ N, there exists a complex algebraic mappingΠ l with values in C N n,l defined for |t|, |τ| < ρ3 and |J l 0− J l 0Id | < ε whichis relatively polynomial with respect to the higher order jets Ji α with |α| ≥l 0 + 1, i = 1, . . ., n, such that for every local holomorphic self-mappingh ∈ H ρ 2,ρ 1M,κ 0 ,ε, the two conjugate relations(6.12){J l h(t) = Π l (t, τ, J l 0+l¯h(τ)),J l¯h(τ) = Πl (τ, t, J l 0+l h(t)).hold for all (t, τ) ∈ M with |t|, |τ| < ρ 3 .Proof. Applying the derivations L β Υ γ to (6.9), and using the chain rule,we obtain(6.13) L β Υ γ h(t) = Π β,γ (t, τ, J l 0+|β|+|γ|¯h(τ)),where the function Π β,γ (as the function Π) is holomorphic for |t|, |τ| < ρ 3and |J l 0−J l 0Id | < ε and relatively polynomial with respect to the jets Jα i with|α| ≥ l 0 + 1. Applying (6.11), we obtain the function Π l , which completesthe proof.6.3. Substitutions of reflection identities. Let π t (t, τ) := t and π τ (t, τ) :=τ denote the two canonical projections. We write h c (t, τ) := (h(t), ¯h(τ)).We make the following slight abuse of notation: instead of rigorously writingh(π t (t, τ)), we write h(t, τ) = h(t) and ¯h(t, τ) = ¯h(τ).Let x ∈ C ν and let Q(x) = (Q 1 (x), . . .,Q 2n (x)) ∈ C{x} 2n . As themultiple flow of L given by (5.3) does not act on the (z, w) variables, wehave the trivial but important property h(L z1(Q(x))) = h(Q(x)). Moregenerally, for every multi-index α ∈ N n , we have ∂ α t h(L z 1(Q(x))) =∂ α t h(Q(x)). Analogously, we have ∂α τ ¯h(L z1 (Q(x))) = ∂ α τ ¯h(Q(x)). Since

for k even, we have Γ k (z (k) ) = L zk(Γ k−1 (z (k−1) )), the following two propertieshold:127(6.14){J l h(Γ k (z (k) )) = J l h(Γ k−1 (z (k−1) )), if k is even;J l¯h(Γk (z (k) )) = J l¯h(Γk−1 (z (k−1) )),if k is odd.Let now κ 0 := l 0 (µ 0 + 1) be the product of the Levi type with the Segretype of M plus 1 and consider the open subset of the κ 0 -order jet spaceC Nn,κ 0 defined by the inequality |Jκ 0− J κ 0Id | < ε. Let z (k) ∈ ∆ mk as in §5.6above. Since the maps Γ k are holomorphic and satisfy Γ k (0) = 0, we maychoose δ > 0 sufficiently small in order that the following two conditionsare satisfied for every k ≤ µ 0 and for and for every |z (k) | < δ:(6.15) |Γ k (z (k) )| < ρ 3 and |J κ 0h(Γ k (z (k) )) − J κ 0Id | < ε.This choice of δ is convenient to make several susbtitutions by means offormulas (6.12). The formulas (6.16) that we will obtain below stronglydiffer from the previous formulas (6.12), because they depend on the jet ofh at the origin only.Lemma 6.3. Shrinking ε if necessary, for every integer k ≤ µ 0 + 1 andfor every integer l ≥ 0, there exists a complex algebraic mapping Π l,k withvalues in C N n,l defined for |t|, |τ| < ρ3 and for |J kl 0− J kl 0Id| < ε, whichis relatively polynomial with respect to the higher order jets Ji α with |α| ≥kl 0 +1, i = 1, . . .,n, and which depends only on the defining functions ξ j −Θ j (ζ, t) of M , such that the following two families of conjugate identitiesare satisfied(6.16){J l h(Γ k (z (k) )) = Π l,k (Γ k (z (k) ), J kl 0+l¯h(0)), if k is odd;J l¯h(Γk (z (k) )) = Π l,k (Γ k (z (k) ), J kl 0+l¯h(0)),if k is even.Proof. For k = 1, replacing (t, τ) by Γ 1 (z (1) ) in the first relation (6.12) andusing the second property (6.14), we get(6.17){J l h(Γ 1 (z (1) )) = Π l (Γ 1 (z (1) ), J l 0+l¯h(Γ1 (z (1) ))) == Π l (Γ 1 (z (1) ), J l 0+l¯h(0)),so the lemma holds true for k = 1 if we simply choose Π l,1 := Π l . Byinduction, suppose that the lemma holds true for k ≤ µ 0 . To fix the ideas,let us assume that this k is even (the odd case is completely similar). Thenreplacing the arguments (t, τ) in the first relation (6.12) by Γ k+1 (z (k+1) ),using again the second property (6.14), and using the induction assumption,namely using the conjugate of the second relation (6.16) with l replaced by

128l 0 + l, we get(6.18) ⎧J l h(Γ k+1 (z (k+1) )) = Π l (Γ k+1 (z (k+1) ), J l 0+l¯h(Γk+1 (z (k+1) ))) =⎪⎨= Π l (Γ k+1 (z (k+1) ), J l 0+l¯h(Γk (z (k) ))) =⎪⎩= Π l (Γ k+1 (z (k+1) ), Π l0 +l,k(Γ k (z (k) ), J kl 0+l 0 +l¯h(0))) =:=: Π l,k+1 (Γ k+1 (z (k+1) ), J (k+1)l 0+l¯h(0)),which yields the desired formula at level k+1. For the above formal compositionformulas to be correct, we possibly have to shrink ε. Finally, a directinspection of relative polynomialness shows that Π l,k+1 is polynomial withrespect to the jet variables J α i with |α| ≥ (k + 1)l 0 + 1, i = 1, . . .,n. Theproof of Lemma 6.21 is complete.6.4. Algebraic parameterization of CR mappings by their jet at the origin.Finally, as in the paragraph after Theorem 5.2, we choose ρ 4 > 0 sufficientlysmall such that Γ µ0 maps the polydisc ∆ mµ0 (η) submersively ontoan open neighborhood of the origin in M which contains the open subsetM ∩ (∆ n (ρ 4 ) × ∆ n (ρ 4 )). From the relation Γ µ0 +1(z (µ0 ), 0) ≡ Γ µ0 (z (µ0 )),it follows trivially that Γ µ0 +1 also induces a submersion from ∆ m(µ0 +1)(η)onto M ∩(∆ n (ρ 4 )×∆ n (ρ 4 )). It follows that the composition π t ◦Γ µ0 +1 alsomaps submersively the polydisc ∆ m(µ0 +1)(η) onto an open neighborhood ofthe origin in C n which contains ∆ n (ρ 4 ). Consequently, in the representationobtained in Lemma 6.21 with l = 0 and k := µ 0 + 1 = 2ν 0 + 2 (which iseven), namely in the representation(6.19) ¯h(Γµ0 +1(z (µ0 +1))) = Π 0,µ0 +1(Γ µ0 +1(z (µ0 +1)), J (µ 0+1)l 0¯h(0)),we can write an arbitrary t ∈ ∆ n (ρ 4 ) in the form Γ µ0 +1(z (µ0 +1)), and finally,conjugating (6.19), we obtain a complex algebraic mapping H withthe property that h(t) = H(t, J (µ 0+1)l 0h(0)). We may now summarize whatwe have proved so far.Theorem 6.4. Let M be a real algebraic generic submanifold in C n passingthrough the origin, of codimension d ≥ 1 and of CR dimension m = n−d ≥1. Assume that M is l 0 -nondegenerate at 0. Assume that M is minimal at0, let ν 0 be the Segre type of M at 0 and let µ 0 := 2ν 0 + 1 be the Segretype of M at 0. Let κ 0 := (µ 0 + 1)l 0 . Let t = (z, w) ∈ C m × C d beholomorphic coordinates vanishing at 0 with T 0 M = {Im w = 0} andlet ρ 1 > 0 be such that M is represented by the complex analytic definingequations ξ j = Θ j (ζ, t), j = 1, . . ., d in ∆ n (ρ 1 ). Then there exist ε > 0,ρ 4 > 0 and there exists a complex algebraic C n -valued mapping H(t, J κ 0)defined for |t| < ρ 4 and for |J κ 0− J κ 0Id | < ε which satisfies H(t, Jκ 0Id ) ≡ tand which depends only on the defining functions ¯w j − Θ j (¯z, t) of M , such

that for every local holomorphic self-mapping h of M belonging to H ρ 2,ρ 1M,κ 0 ,ε ,we have the representation formula(6.20) h(t) = H(t, J κ 0h(0)),for all t ∈ C n with |t| < ρ 4 . Furthermore the mapping H depends neither onthe choice of smaller radii ˜ρ 1 ≤ ρ 1 , ˜ρ 2 ≤ ρ 2 , ˜ρ 3 ≤ ρ 3 and ˜ρ 4 ≤ ρ 4 satisfying0 < ˜ρ 4 < ˜ρ 3 < ˜ρ 2 < ˜ρ 1 nor on the choice of a smaller constant ˜ε < ε,so that the first sentence of property (3) in Theorem 4.1 holds true. Finally,if M is real analytic, the same statement holds with the word “algebraic”everywhere replaced by the word “analytic”.It remains now to construct the submanifold E whose existence is statedin Theorem 4.1 and to establish that H ρ 2,ρ 1M,κ 0 ,εmay be endowed with the structureof a local real algebraic Lie group.6.5. Local real algebraic Lie group structure. In order to construct thissubmanifold E, we introduce the κ 0 -th jet mapping J κ 0: H ρ 2,ρ 1M,κ 0 ,ε →C Nn,κ 0 defined by Jκ 0(h) := (∂t α h(0)) |α|≤κ0 = J κ 0h(0). The followinglemma is crucial.Lemma 6.5. Shrinking ε if necessary, the set(6.21) E := J κ 0(H ρ 2,ρ 1M,κ 0 ,ε ) = {Jκ 0h(0) : h ∈ H ρ 2,ρ 1M,κ 0 ,ε }is a real algebraic totally real submanifold of the polydisc {J κ 0∈ C Nn,κ 0 :|J κ 0− J κ 0Id | < ε}.Proof. Let h ∈ H ρ 2,ρ 1M,κ 0 ,ε. Substituting the representation formula h(t) =H(t, J κ 0h(0)) given by Theorem 6.4 in the defining equations of M, we get(6.22) r j (H(t, J κ 0h(0)), H(τ, J κ 0¯h(0))) = 0,for j = 1, . . .,d and (t, τ) ∈ M with |t|, |τ| < ρ 4 . As (t, τ) ∈ M , wereplace ξ by Θ(ζ, t) and we use the 2m + d coordinates (t, ζ) on M . So, byexpanding the functions (6.22) in power series with respect to (t, ζ), we canwrite(6.23)∑r j (H(t, J κ 0), H(ζ, Θ(ζ, t), J κ 0 )) = t α ζ β C j,α,β (J κ 0, J κ 0 ).α∈N n , β∈N mHere, we obtain an infinite collection of complex-valued real algebraic functionsC j,α,β defined in {|J κ 0− J κ 0Id| < ε} with the property that a mappingH(t, J κ 0) sends M ∩ ∆ n (ρ 4 ) into M if and only if(6.24) C j,α,β (J κ 0, J κ 0 ) = 0, ∀ j, α, β.Consequently, the set E defined by the vanishing of all the equations (6.24)is a real algebraic subset.129

130It follows from the representation formula (6.20) that the mapping J κ 0is injective and from the Cauchy integral formula that J κ 0is continuouson its domain of definition H ρ 2,ρ 1M,κ 0 ,εendowed with the topology of uniformconvergence on compact sets.On the reverse side, let J κ 0∈ E. Then the mapping h(t) := H(t, J κ 0)defined for |t| < ρ 4 maps M ∩ ∆ n (ρ 4 ) into M. Applying Theorem 6.4to this mapping h(t), with ρ 1 replaced by ρ 4 , we deduce that there existsa radius ρ 6 < ρ 4 such that we can represent h(t) = H(t, J κ 0h(0)) for|t| < ρ 6 , with the same mapping H, as stated in the end of Theorem 6.4.By differentiating this representation with respect to t at t = 0, we deducethat J κ 0h(0) = ([∂t αH(t, Jκ 0h(0))] t=0 ) |α|≤κ0 . Consequently, sinceh(t) = H(t, J κ 0) by definition, we get J κ 0= ([∂t αH(t, Jκ 0)] t=0 ) |α|≤κ0 . Inconclusion, we proved that J κ 0(H(t, J κ 0)) = J κ 0for every J κ 0∈ E, soJ κ 0has a continuous local inverse on E, formally defined by H(t, J κ 0).It follows from the above two paragraphs that the mapping J κ 0is a localhomeomorphism from a neighborhood of the identity in H ρ 2,ρ 1M,κ 0 ,εonto itsimage E.Furthermore, we claim that the real algebraic subset E is in fact geometricallysmooth at every point, namely it is a real algebraic submanifold.Indeed, let J κ 01 be a regular point of E where E is of maximal geometricaldimension c 0 , with J κ 01 arbitrarily close to the identity jet J κ 0Id . Leth 1 ∈ H ρ 2,ρ 1M,κ 0 ,ε such that Jκ 01 = J κ 0(h 1 ). Let U 1 be a small neighborhoodof J κ 01 in C Nn,κ 0 in which E ∩ U1 is a regular c 0 -dimensional real algebraicsubmanifold and consider the complex algebraic mapping defined over U 1by(6.25) F 1 (J κ 0) := ([∂ α t (h −11 (H(t, J κ 0)))] t=0 ) |α|≤κ0 ∈ C Nn,κ 0 .We have F 1 (J κ 01 ) = Jκ 0Id and the restriction of F 1 to E∩U 1 induces a homeomorphismonto its image, which is a neighborhood of J κ 0Idin E. We remindthat the mapping J κ 0→ ([∂t α (H(t, J κ 0))] t=0 ) |α|≤κ0 restricted to E ∩ U 1 isthe identity and consequently of constant rank equal to c 0 . As h 1 is invertible,it follows from the chain rule by developing (6.25) that F 1 | E∩U1 is alsoof locally constant rank equal to c 0 . This proves that E is a c 0 -dimensionalreal algebraic submanifold in C Nn,κ 0 through Jκ 0Id. More generally, this reasoningshows that E is geometrically smooth at every point.Finally, applying Lemma 6.3 with the odd integer k = µ 0 = 2ν 0 + 1(instead of k = µ 0 + 1), we get a new, different representation formulah(t) = ˜H(t, J l 0µ 0¯h(0)) (notice ¯h(0)). Accordingly, we can define a realalgebraic submanifold Ẽ. It is clear that we can identify E and Ẽ, since theyboth parametrize the local biholomorphic self-mappings of M, so they arealgebraically equivalent by means of the natural projection from the l 0 (µ 0 +1)-th jet space onto the l 0 µ 0 -th jet space. Next, we see by differentiating

h(t) = ˜H(t, J l 0µ 0¯h(0)) with respect to t that(6.26) J l 0µ 0h(0) = ([∂tα ˜H(t, J l 0µ 0¯h(0))]t=0 ) |α|≤l0 µ 0.Consequently, if K is the holomorphic map defined by(6.27) K(J l 0µ 0) := ([∂ α t˜H(t, J l 0µ 0] t=0 ) |α|≤l0 µ 0),we get the equality J l 0µ 0= K(J l 0µ 0) for every Jl 0 µ 0∈ Ẽ, which provesthat Ẽ is totally real. It follows that E is totally real, which completes theproof.Lemma 6.6. The submanifold E is naturally equipped with a local realalgebraic Lie group structure in a neighborhood of J κ 0Id .Proof. Indeed, let us parametrize E by a real algebraic mapping(6.28) R c 0∋ (e 1 , . . .,e c0 ) ↦−→ j κ0 (e) ∈ C Nn,κ 0 ,where c 0 is the dimension of E. Here, to avoid excessive formal complexity,we shall avoid to mention all the polydiscs of variation of the variables. Fore ∈ E, we shall use the notation(6.29) H(t; e) := H(t, j κ0 (e)).Let e ∈ E and e ′ ∈ E, set J κ 0:= j κ0 (e) and ′ J κ 0:= j κ0 (e ′ ). Then we candefine the Lie group multiplication µ J by(6.30) µ J ( ′ J κ 0, J κ 0) := ([∂ α t (H(H(t, Jκ 0), ′ J κ 0))] t=0 ) |α|≤κ0 .Accordingly, in terms of the coordinates (e 1 , . . .,e c0 ) on E, the Lie groupmultiplication µ is defined by(6.31) µ(e, e ′ ) := (j κ0 ) −1 (µ J (j κ0 (e ′ ), j κ0 (e))) ∈ R c 0It follows from the algebraicity of the mappings H and j κ0 that the mappingsµ J and µ are algebraic.We must check the associativity of µ, namely µ(µ(e, e ′ ), e ′′ ) =µ(e, µ(e ′ , e ′′ )). So we set h(t) := H(t, j κ0 (e)), h ′ (t) := H(t, j κ0 (e ′ ))and h ′′ (t) := H(t, j κ0 (e ′′ )). By the definition (6.30), we haveµ J (j κ0 (e), j κ0 (e ′ )) = J κ 0(h ◦ h ′ )(0). Applying then Theorem 6.4, weget H(t, J κ 0(h ◦ h ′ )(0)) ≡ (h ◦ h ′ )(t). Consequently, using again (6.30)and the associativity of the composition of mappings, we may compute(6.32) ⎧µ J (µ J (j κ0 (e), j κ0 (e ′ )), j κ0 (e ′′ )) = µ J (J κ 0((h ◦ h ′ )(0), j κ0 (e ′′ ))⎪⎨⎪⎩= J κ 0((h ◦ h ′ ) ◦ h ′′ )(0)= J κ 0(h ◦ (h ′ ◦ h ′′ ))(0)= µ J (j κ0 (e), J κ 0(h ′ ◦ h ′′ )(0))= µ J (j κ0 (e), µ J (j κ0 (e), j κ0 (e ′ ))),131

132which proves the associativity.Finally, we may define an algebraic inversion mapping ι as follows.First of all, for J κ 0close to J κ 0Id , the mapping h(t) := H(t, Jκ 0) =t + ∑ α∈Nt α H n α (J κ 0) is an invertible algebraic biholomorphic mapping.Here, the coefficients H α (J κ 0) are algebraic functions of J κ 0which vanishat J κ 0Id (since H(t, Jκ 0Id) ≡ t in Theorem 6.4). From the algebraic implicitfunction theorem, it follows that the local inverse h −1 (t) writes uniquely inthe form h −1 (t) = t+ ∑ α∈Nt α ˜Hα (J κ 0) =: ˜H(t, J κ 0), where the ˜H n α (J κ 0)are algebraic functions of J κ 0also satisfying ˜H α (J κ 0Id) = 0. Consequently,choosing e ∈ E such that J κ 0= j κ0 (e), we can define(6.33) ι J (J κ 0) := ([∂ α t˜H(t, J κ 0)] t=0 ) |α|≤κ0 .Accordingly, in terms of the coordinates (e 1 , . . .,e c0 ) on E, the Lie groupinverse mapping is defined by(6.34) ι(e) := (j κ 0) −1 (i J (j κ0 (e))).Of course, with this definition we have ι J (J κ 0Id ) = Jκ 0Id. Finally, we leave tothe reader to verify that µ J (j κ0 (e), i J (j κ0 (e))) = J κ 0Id. This completes theproof of property (4) of Theorem 4.1.End of proof of Theorem 4.1. We notice that statement (5) does not needto be proved. Furthermore that the dimensional inequality c 0 ≤ (n+κ 0)!n! κ 0in!(6) follows from the fact each local biholomorphic mapping in the localLie group H ρ 2,ρ 1M,κ 0 ,ε ∼ = E writes uniquely as h(t) = H(t, J κ 0h(0)), so thecomplex dimension of the local Lie group E is ≤ (n+κ 0)!n! κ 0 !, the dimensionof the κ 0 -th jet space. As E is totally real, the real dimension of E is also≤ (n+κ 0)!n! κ 0 !. Finally, it follows that the real local Lie algebra of vector fieldsHol(M, ∆ n (ρ 5 )) is of dimension ≤ (n+κ 0)!n! κ 0 !. The proof of Theorem 4.1 iscomplete.§7. DESCRIPTION OF EXPLICIT FAMILIES OF STRONG TUBES IN C n7.1. Introduction. Theorems 1.1, 1.4 and 1.5 provide sufficient conditionsfor some real analytic real submanifold in C n to be not locally algebraizable.For the sake of completeness, we exhibit explicit examples of such nonalgebraizablesubmanifolds which are effectively strong tubes and effectivelynonalgebraizable, proving corollaries 1.2, 1.3, 1.6 and 1.7. Consequently wewill deal with the two following families of nonalgebraizable real analyticLevi nondegenerate hypersurfaces in C n (n ≥ 2) : the Levi nondegeneratestrong tube hypersurfaces in C n and the strongly rigid hypersurfaces in C n .For heuristic reasons, we shall sometimes start with the case n = 2 and treatthe general case n ≥ 2 afterwards. In fact, our goal will be to construct

infinite families of pairwise non biholomorphically equivalent and non locallyalgebraizable hypersurfaces. Our computations for the construction offamilies of manifolds with a control on the structure of their automorphismgroup are all based on the Lie theory of symmetries of differential equations.For the convenience of the reader, we recall briefly the procedure (see[Su2001a,b], [GM2001a,b,c] for more details).7.2. Hypersurfaces and differential equations. Let M be a real analytichypersurface in C n . Assume that M is Levi nondegenerate at one of itspoints p. Then there exist some local holomorphic coordinates (z, w) =(z, u + iv) ∈ C n−1 × C vanishing at p such that M is given by the realanalytic equation(7.1) v = ϕ(z, ¯z, u) = ε 1 |z 1 | 2 + · · · + ε n−1 |z n−1 | 2 + ψ(z, ¯z, u),where ε k = ±1, k = 1, . . ., n − 1 and where ψ = O(3). Passing to theextrinsic complexification M of M, we may consider the variables ¯z and¯w as independent complex parameters ζ ∈ C n−1 and ξ ∈ C. Then theassociated complex defining equation is of the form(7.2) w = Θ(z, ζ, ξ) = ξ + 2i(ε 1 z 1 ζ 1 + · · · + ε n−1 z n−1 ζ n−1 + Ξ(z, ζ, ξ)),where Ξ = O(3). By [Me1998] (cf. §5.1 above), for τ p = (ζ p , ξ p ) fixed, thefamily of complexified Segre varieties S τp := {(t, τ p ) : w = Θ(z, τ p )} isinvariantly and biholomorphically attached to M.Following [Se1931] and [Su2001a,b], we may consider this family as afamily of graphs of the solutions of a second order completely integrablesystem of partial differential equations as follows. By differentiating the leftand the right hand sides of (7.2) with respect to z k , we get(7.3) ∂ zk w = ∂ zk Θ(z, τ) = 2i(ε k ζ k + ∂ zk Ξ(z, τ)),for k = 1, . . .,n − 1. Here, we consider w as a function of z. Usingthe analytic implicit function theorem to solve τ in the 1 + (n − 1) = nequations (7.2) and (7.3), we may express τ in terms of w, of z and of thefirst order derivative w zl , which yields(7.4) τ = Π(z, w, (∂ zl w) 1≤l≤n−1 ),where Π is holomorphic in its variables. If we take the second derivativew zk1 z k2of w and replace the value of τ, we get the desired system of partialdifferential equations:(7.5)∂ 2 z k1 z k2w = ∂ 2 z k1 z k2Θ(z, τ) = ∂ 2 z k1 z k2Θ(z, Π(z, w, (∂ zl w) 1≤l≤n−1 )) =:=: F k1 ,k 2(z, w, (∂ zl w) 1≤l≤n−1 ).133

134Here, k 1 , k 2 = 1, . . .,n − 1 and the F k1 ,k 2≡ F k2 ,k 1are holomorphic intheir variables. We denote by E M this system of partial differential equations(here, to construct E M , we have used the Levi nondegeneracy of M butwe note that if M were finitely nondegenerate the same conclusion wouldbe true, by considering some derivatives of w of larger order). Since the solutionsof E M are precisely the complexified Segre varieties S τ , the systemE M is completely integrable.To study the local geometry of M, we may consider on one hand thereal Lie algebra of infinitesimal CR automorphisms of M (cf. §2.2), namelyAut CR (M) = 2 ReHol(M). On the other hand, following the general ideasof Lie (cf. the modern restitution by Olver in [Ol1986, Ch 2]), we may considerthe Lie algebra of infinitesimal generators of the local symmetry groupof the system of partial differential equations E M , which we shall denote bySym(E M ). By definition, Sym(E M ) consists of holomorphic vector fieldsin the (z, w)-space whose local flow transforms the graph of every solutionof E M (namely a complexified Segre variety) into the graph of another solutionof E M (namely into another complexified Segre variety). The linkbetween Aut CR (M) and Sym(E M ) is as follows: by [Ca1932, p. 30–32],one can prove that Aut CR (M) is a maximally real subspace of Sym(E M )(see also [Su2001a,b], [GM2001a,b,c]).The computation of explicit generators of Sym(E M ) may be performedusing the Lie theory of symmetries of differential equations. By inspectingsome examples, it appears that dealing with Sym(E M ) generally shortensthe complexity of the computation of Aut CR (M) by at least one half.The Lie procedure to compute Sym(E M ) is as follows. Let Jn−1,1 2 (C)denote the space of second order jets of a function w(z 1 , . . ., z n−1 )of (n − 1) complex variables, equipped with independent coordinates(z, w, Wl 1,W k 2 1 ,k 2) corresponding to (z, w, w zl , w zk1 z k2), wherel = 1, . . .,n − 1, where k 1 , k 2 = 1, . . ., n − 1, and where we of courseidentify Wk 2 1 ,k 2with Wk 2 2 ,k 1. To the system E M , we associate the complexsubmanifold of Jn−1,1(C) 2 defined by replacing the derivatives of w by theindependent jet variables in the system E M , which yields (cf. (7.5)):(7.6) W 2 k 1 ,k 2= F k1 ,k 2(z, w, (W 1l ) 1≤l≤n−1 ),for k 1 , k 2 = 1, . . .,n −1. Let ∆ M denote this submanifold. By Lie’s theory,every vector field X = ∑ n−1k=1 Qk (z, w) ∂ zk +R(z, w) ∂ w defined in a neighborhoodof the origin in C n can be uniquely lifted to a vector field X (2) inJn−1,1 2 (C), which is called the second prolongation of X (by definition, thelift X (2) shows how the flow of X transforms second order jets of graphs of

functions w(z)). The coefficients R 1 l and R 2 k 1 ,k 2of the second prolongation135(7.7) X (2) =∑n−1k=1Q k∂+R ∂n−1∂z k ∂w + ∑l=1R 1 l∂∂W 1l+∑n−1k 1 ,k 2 =1R 2 k 1 ,k 2∂∂W 2 k 1 ,k 2,are completely determined by the following universal formulas(cf. [Ol1986], [Su2001a,b], [GM2001a]):(7.8) ⎧⎪⎨⎪⎩R 1 l = ∂ zl R + ∑ m 1[δ m 1l∂ w R − ∂ zl Q m 1] W 1 m 1+ ∑[−δ m 1l∂ w Q m 2] Wm 1 1Wm 1 2.m 1 ,m 2]R 2 k 1 ,k 2= ∂ 2 z k1 z k2R + ∑ m 1[δ m 1k 1∂ 2 z k2 w R + δm 1k 2∂ 2 z k1 w R − ∂2 z k1 z k2Q m 1W 1 m 1++ ∑ []δ m 1,m 2k 1 , k 2∂ 2 w 2R − δm 1k 1∂z 2 k2 w Qm 2− δ m 1k 2∂z 2 k1 w Qm 2Wm 1 1Wm 1 2+m 1 ,m 2+ ∑ [−δm 1 ,m 2k 1 , k 2∂ ] 2 w 2Qm 3Wm 1 1Wm 1 2Wm 1 3+m 1 ,m 2 ,m 3+ ∑ []δ m 1,m 2k 1 , k 2∂ w R − δ m 1k 1∂ zk2 Q m 2− δ m 1k 2∂ zk1 Q m 2Wm 2 1 ,m 2+m 1 ,m 2+ ∑ [−δm 1 ,m 2k 1 , k 2∂ w Q m 3− δ m 2,m 3k 1 , k 2∂ w Q m 1− δ m 3,m 1k 1 , k 2∂ w Q ] m 2Wm 1 1Wm 2 2 ,m 3.m 1 ,m 2 ,m 3In these formulas, by δl m we denote the Kronecker symbol equal to 1 if l =m and to 0 otherwise. The multiple Kronecker symbol δ m 1,m 2l 1 , l 2is defined tobe the product δ m 1l 1·δ m 2l 2. Finally, in the sums ∑ m 1, ∑ m 1 ,m 2and ∑ m 1 ,m 2 ,m 3,the integers m 1 , m 2 , m 3 run from 1 to n − 1. We would like to mentionthat in [GM2001a], we also provide some explicit expression of the k-thprolongation X (k) for k ≥ 3.Then the Lie criterion states that a holomorphic vector field X belongsto Sym(E M ) if and only if its second prolongation X (2) is tangent to ∆ M([Ol1986, Ch 2]). This gives the following equations:(7.9) R 2 k 1 ,k 2−∑n−1k=1Q k ∂ zk F k1 ,k 2− R ∂ w F k1 ,k 2−∑n−1l=1R 1 l ∂ W 1lF k1 ,k 2≡ 0,where 1 ≤ k 1 , k 2 ≤ n −1 and where each occurence of Wl 21 ,l 2is replaced byits value F l1 ,l 2on ∆ M . By developping (7.9) in power series with respect tothe variables Wl 1 , we get an expression of the form(7.10)∑l 1 ,...,l n−1 ≥0W 1l 1 · · ·W 1l n−1Φ l1 ,...,l n−1≡ 0,

136where each term Φ l1 ,...,l n−1is a certain linear partial differential expressioninvolving the derivatives of Q 1 , . . .,Q n−1 , R up to order two with coefficientsbeing holomorphic functions of (z, w). The determination of a systemof generators X 1 , . . .,X c of Sym(E M ) is obtained by solving the infinitecollection of these linear partial differential equations Φ l1 ,...,l n−1= 0(cf. [Ol1986], [Su2001a,b], [GM2001a,b,c]). We shall apply this generalprocedure to provide different families of nonalgebraizable real analytic hypersurfacesin C n .7.3. Hypersurfaces in C 2 with control of their CR automorphism group.The goal of this paragraph is to construct some classes of strong tubes,namely tubes having the smallest possible CR automorphism group. Westart with the case n = 2 and study afterwards the case n ≥ 3 in the nextsubparagraph. Let M χ be the strong tube hypersurface in C 2 defined by theequation(7.11) M χ : v = ϕ(y) := y 2 + y 6 + y 9 + y 10 χ(y).where χ is a real analytic function defined in a neighborhood of the originin R.Lemma 7.1. The hypersurfaces M χ are pairwise not biholomorphicallyequivalent strong tubes.Proof. To check that M χ is a strong tube, it suffices to show that every hypersurfaceof the form v = y 2 + y 6 + O(y 9 ) is a strong tube (the term y 9will be used afterwards). Writing v = (w − ¯w)/2i and y = (z − ¯z)/2i,considering w as a function of z and ¯w, ¯z as constants, the differentiation ofw with respect to z in (7.11) yields:(7.12) ∂ z w = 2y + 6y 5 + O(y 8 ).The implicit function theorem yields:(7.13) y = (1/2)∂ z w − (3/2 5 )(∂ z w) 5 + O((∂ z w) 8 ).One further differentiation of equation (7.12) with respect to z gives:(7.14) ∂ 2 zz w = −i − (15i) y4 + O(y 7 ).Replacing y in this equation by its value obtained in (7.13), we obtain thefollowing second order ordinary equation E M satisfied by ∂ z w and ∂ 2 zzw:(7.15) ∂ 2 zzw = −i − (15i/2 4 )(∂ z w) 4 + O((∂ z w) 7 ).In the four dimensional jet space J1,1 2 (C) equipped with the coordinates(z, w, W 1 , W 2 ) the equation of the corresponding complex hypersurface∆ M is of course:(7.16) W 2 = −i − (15i/2 4 )(W 1 ) 4 + O((W 1 ) 7 ).

Then the Lie criterion states that a holomorphic vector field X = Q ∂ z +R ∂ w belongs to Sym(E M ) if and only if its second prolongation X (2) =Q ∂ z + R ∂ w + R 1 ∂ W 1 + R 2 ∂ W 2 is tangent to ∆ M , where the coefficientsR 1 and R 2 are given by the formulas (7.8) specified for n = 2, namely:(7.17) ⎧⎪⎨R 1 = ∂ z R + [∂ w R − ∂ z Q] W 1 − ∂ w Q (W 1 ) 2 .R 2 = ∂zz 2 ⎪⎩R + [2∂2 zw R − ∂2 zz Q] W 1 + [∂ww 2 R − 2∂2 zw Q] (W 1 ) 2 − ∂ww 2 Q (W 1 ) 3 ++ [∂ w R − 2∂ z Q] W 2 − 3∂ w Q W 1 W 2 .The tangency condition yields the following equation which is satisfied on∆ M , i.e. after replacing W 2 by its value given by (7.16):(7.18) R 2 + (15i/2 2 )R 1 (W 1 ) 3 + O((W 1 ) 6 ) = 0.By expanding equation (7.18) in powers of W 1 up to order five, we obtainthe following system of six linear partial differential equations which mustbe satisfied by the derivatives of Q and R up to order two:⎧(e 0 ) : ∂zz 2 R − i(∂ wR − 2∂ z Q) ≡ 0.(e 1 ) : 2∂ 2 zw R − ∂2 zz Q ≡ 0.137(7.19)⎪⎨(e 2 ) : ∂ 2 ww R − 2∂2 zw Q ≡ 0.(e 3 ) : −∂ 2 ww Q + 15i2 2 ∂ z R ≡ 0.⎪⎩(e 4 ) : − 15i2 4 (∂ w R − 2∂ z Q) + 15i2 2 (∂ w R − ∂ z Q) ≡ 0.(e 5 ) : − 15i2 4 (−3∂ w Q) − 15i2 2 (∂ w Q) ≡ 0.It follows from the equation (e 5 ) that ∂ w Q ≡ 0 which implies ∂ 2 ww Q ≡ 0.Then by equation (e 3 ) we obtain ∂ z R ≡ 0, implying ∂ 2 zzR ≡ 0. From equation(e 0 ) we get ∂ w R ≡ 2∂ z Q and, from equation (e 4 ), we get ∂ w R ≡ ∂ z Q.Consequently ∂ z R ≡ ∂ w R ≡ ∂ z Q ≡ ∂ w Q ≡ 0. Since the two vector fields∂ z and ∂ w evidently belong to Sym(E M ), it follows that dim C Sym(E M ) =2. Finally, this implies that dim R Aut CR (M) = 2 and that Aut CR (M) isgenerated by ∂ w + ∂ ¯w and ∂ z + ∂¯z .Next, let χ(y) and χ ′ (y ′ ) be two real analytic functions, and assume thatM χ and M ′ χ ′ are biholomorphically equivalent. Let t′ = h(t) be such anequivalence. Reasoning as in §4 and taking into account that both are strongtubes, we see that h ∗ (∂ z ) and h ∗ (∂ w ) must be linear combinations of ∂ z ′ and∂ w ′ with real coefficients. It follows that h must be linear, of the form z ′ =az +bw, w ′ = cz +dw, where a, b, c and d are real. Since T 0 M χ = {v = 0}

138and T 0 M χ ′ = ′ {v′ = 0}, we have c = 0. Next, in the equation(7.20) ⎧⎪⎨d(y 2 + y 6 + y 9 + y 10 χ(y)) ≡ [ay + b(y 2 + y 6 + y 9 + y 10 χ(y))] 2 ++ [ay + b(y 2 + y 6 + y 9 + y 10 χ(y))] 6 + [ay + b(y 2 + y 6 + y 9 + y 10 χ(y))] 9 +⎪⎩+ [ay + b(y 2 + y 6 + y 9 + y 10 χ(y))] 10 χ ′ (ay + b(y 2 + y 6 + y 9 + y 10 χ(y))),we firstly see that b = 0, and then from(7.21) d(y 2 + y 6 + y 9 + y 10 χ(y)) ≡ a 2 y 2 + a 6 y 6 + a 9 y 9 + a 10 y 10 χ ′ (ay),we see that a = d = 1. In other words, h = Id, whence y ′χ ′ (y ′ ) ≡ χ(y). This proves Lemma 7.1.= y andIn the remainder of §7, we shall exhibit other classes of hypersurfaceswith a control on their CR automorphism group. Since the computations aregenerally similar, we shall summarize them.7.4. Some classes of strong tube hypersurfaces in C n . GeneralizingLemma 7.1, we may state:Lemma 7.2. The real analytic hypersurfaces M χ1 ,...,χ n−1⊂ C n of equation∑n−1(7.22) v =k=1[ε k y 2 k + y 6 k + y 9 ky 1 · · ·y k−1 + y n+8kχ k (y 1 , . . ., y n−1 )],where ε k = ±1, are pairwise not biholomorphically equivalent strong tubes.Proof. The associated system of partial differential equations is of the form(7.23) {∂2zk z kw = − iε k − (15i/2 4 ) (∂ zk w) 4 + O((∂ z1 w) 7 ) + · · · + O((∂ zn−1 w) 7 ),∂ 2 z k1 z k2w = 0, for k 1 ≠ k 2 .Using the formulas (7.8) and inspecting the coefficients of the monomials inthe Wl 1 up to order five in the (n −1) equations extracted from the set of Lieequations(7.24) {R2k,k + (15i/2 2 )(Wk 2 ) Rk 1 + O((W1 1 ) 6 ) + · · · + O((Wn−1) 1 6 ) = 0,R 2 k 1 ,k 2= 0, for k 1 ≠ k 2 ,we get ∂ zl R ≡ ∂ w R ≡ ∂ zl Q k ≡ ∂ w Q k ≡ 0 for l, k = 1, . . .,n − 1. Thus,M χ1 ,...,χ n−1is a strong tube.Next, reasoning as in the end of the proof of Lemma 7.1, we see firstthat an equivalence between M χ1 ,...,χ n−1and M ′ χ ′ 1 ,...,χ′ n−1must be of the formz k ′ = ∑ n−1l=1 λl k z l, w ′ = µ w, where λ l k , 1 ≤ l, k ≤ n − 1 and µ are real.Inspecting the terms of degree 9, 10, . . ., n + 7, we get λ l k = 0 if k ≠ l, i.e.

y k ′ = λk k y k and w ′ = µ w. Finally, λ k k = 1 and µ = 1, which completes theproof.7.5. Families of strongly rigid hypersurfaces. Alongside the same recipe,we can study some classes of hypersurfaces of the form v = ϕ(z¯z).Lemma 7.3. The Lie algebra Hol(M χ ) of the rigid real analytic hypersurfacesM χ in C 2 of equation v = ϕ(z¯z) = z¯z + z 5¯z 5 + z 7¯z 7 + z 8¯z 8 χ(z¯z)is two-dimensional and generated by ∂ w and iz∂ z . Furthermore, M χ is biholomorphicallyequivalent to M ′ χ ′ if and only if χ = χ′ .Proof. The associated differential equation is of the form(7.25) ∂ 2 zz w = [5z3 /4](∂ z w) 5 − [21z 5 /32](∂ z w) 7 + O((∂ z w) 9 ).Extracting from the associated Lie equations (7.10) the coefficients of themonomials (W 1 ) 4 , (W 1 ) 5 , (W 1 ) 6 and (W 1 ) 7 , we obtain four equationswhich are solved by z∂ z Q−Q ≡ 0, ∂ w Q ≡ 0, ∂ z R ≡ 0 and ∂ w R ≡ 0. Next,if M χ and M χ ′ ′ are biholomorphically equivalent, reasoning as in §4, takinginto account that h ∗ (iz∂ z ) and h ∗ (∂ w ) are linear combinations of iz ′ ∂ z ′ and∂ w ′ with real coefficients, we see first that z ′ = λ z e γw/2i and w ′ = µ w forsome three real constants γ, λ ≠ 0 and µ ≠ 0. Replacing z ′ and w ′ in theequation of M χ ′ ′, we get γ = 0, µ = 1 and λ ± 1. In other words, z′ = ±zand w ′ = w, which entails χ ′ (z ′¯z ′ ) ≡ χ(z¯z), as claimed.Perturbing this family we may exhibit other strongly rigid hypersurfaces :Lemma 7.4. The Lie algebra Hol(M χ ) of the real analytic hypersurfacesM χ in C 2 of equation v = ϕ(z, ¯z) = z¯z + z 5¯z 5 + z 7¯z 7 + z 8¯z 8 (z + ¯z) +z 10¯z 10 χ(z, ¯z) is one-dimensional and generated by ∂ w . Furthermore, M χ isbiholomorphically equivalent to M ′ χ ′ if and only if χ = χ′ .Proof. We already know that z∂ z Q − Q ≡ ∂ w Q ≡ ∂ z R ≡ ∂ w R ≡ 0.Extracting from the associated Lie equations (7.10) the coefficient of themonomials (W 1 ) 8 , we also get Q ≡ 0. Next, let M χ and M χ ′ ′ be biholomorphicallyequivalent. Let t ′ = h(t) be such an equivalence. Usingh ∗ (∂ w ) = µ ∂ w ′, where µ ∈ R is nonzero, we get z ′ = f(z) andw ′ = µw + g(z). Next from the equation(7.26) {µ(z¯z + z5¯z 5 + z 7¯z 7 + z 8¯z 8 (z + ¯z) + O(z 9¯z 9 )) + [g(z) − ḡ(¯z)]/2i ≡≡ f(z) ¯f(¯z) + f(z) 5 ¯f(¯z) 5 + f(z) 7 ¯f(¯z) 7 + f(z) 8 ¯f(¯z) 8 (f(z) + ¯f(¯z)) + O(z 9¯z 9 ),we get firstly f(z) = √ |µ|e iθ z by differentiating with respect to ¯z at ¯z = 0and secondly µ = e iθ = 1, which completes the proof.We provide a second family of strongly rigid hypersurfaces in C 2 with aone-dimensional Lie algebra :139

140Lemma 7.5. The Lie algebra Hol(M χ ) of the real analytic hypersurfacesM χ ⊂ C 2 of equation v = z¯z + z 5¯z 5 (z + ¯z) + z 10¯z 10 χ(z, ¯z) is onedimensionaland generated by ∂ w . Furthermore M χ is biholomorphicallyequivalent to M ′ χ ′ if and only if χ = χ′ .Proof. The derivatives ∂ z w and ∂ 2 zw of w with respect to z are given by :{ 2 ∂z w = 2i¯z + 12iz 5¯z 5 + 10iz 4¯z 6 + O(¯z 10 )),(7.27)∂ 2 zz w = 60iz4¯z 5 + 40iz 3¯z 6 + O(¯z 10 ).Replacing ¯z in the second equation by its expression given by the first equationwe obtain the following second order differential equation, interpretedin the jet space :(7.28)W 2 = [15z 4 /8] (W 1 ) 5 − [5iz 3 /8] (W 1 ) 6 − [225z 9 /64] (W 1 ) 9 + O((W 1 ) 10 ).Solving the partial differential equations involving Q, ∂ z Q, ∂ w Q, ∂ z R and∂ w R given in the coefficients of (W 1 ) 4 , (W 1 ) 5 , (W 1 ) 6 , (W 1 ) 7 and (W 1 ) 9we obtain Q ≡ ∂ z Q ≡ ∂ w Q ≡ ∂ z R ≡ ∂ w R ≡ 0 which is the desired information.Finally, proceeding exactly as in the end of the proof of Lemma 7.4,we see that the M χ are pairwise biholomorphically not equivalent.The dimension of Hol(M) for the five examples of Corollary 1.3, for theseven examples of Theorem 1.4, for the seven examples of Corollary 1.7and for the hypersurface v = e z¯z − 1 at a point p with z p ≠ 0 was computedwith the package diffalg of Maple Release 6. Since at a pointp with z p ≠ 0 the hypersurface v = e z¯z − 1 is biholomorphically equivalentto the hypersurface M a of equation v = ϕ a (y) := e a(ey−1) − 1with a = |z p | 2 , this defines a strong tube. Applying Theorem 1.1 andLemma 3.3, we see that M a is not locally algebraizable at the origin, becauseϕ a yy (y) = aey e a(ey−1) + a 2 e 2y e a(ey−1) and ϕ a y (y) = aey e a(ey−1) arealgebraically independent. Finally all the examples of Corollary 1.3, Theorem1.4 and Corollary 1.6 are not locally algebraic since they satisfy therequired transcendence conditions.§8. ANALYTICITY VERSUS ALGEBRAICITYIntuitively there seems to be much more analytic mappings, manifoldsand varieties than algebraic ones. Our goal is to elaborate a precise statementabout this. By complexification, every local real analytic object yields a localcomplex analytic object, so we shall only work in the holomorphic category.Let ∆ n be the complex polydisc of radius one in C n and ∆ n its closure.Let k ∈ N. We consider the space O k (∆ n ) := O(∆ n ) ∩ C k (∆ n ) of holomorphicfunctions extending up to the boundary as a function of class C k. This is a Banach space for the C k norm ||ϕ|| k := ∑ kl=0 sup z∈∆ n|ϕ z l(z)|.

The last statements of Corollaries 1.2 and 1.6 are a direct consequence ofthe following lemma.Lemma 8.1. The set of holomorphic functions ϕ ∈ O k (∆ n ) such that thereexists a polynomial P such that(8.1) P(z, j k ϕ(z)) ≡ 0,is of first category, namely it can be represented as the countable union ofnowhere dense closed subsets. Conversely, the set of functions ϕ ∈ O k (∆ n )such that there is no algebraic dependence relation like (8.1) is generic inthe sense of Baire, namely it can be represented as the countable intersectionof everywhere dense open subsets.Proof. Let N ∈ N. Consider the set F N of functions ϕ such that there existsa polynomial of degree N satisfying (8.1). It suffices to show that F Nis closed and that its complement is everywhere dense. Suppose that a sequence(ϕ (m) ) m∈N converges to ϕ ∈ O k (∆ n ). Let the zero-set of a degreeN polynomial P (m)N(z, J k) contain the graph of the k-jet of ϕ (m) . The coefficientsof P (m)Nbelong to a certain complex projective space P A(C), wherethe integer A = A(n, k) is independent of m. By compactness of P A (C),passing to a subsequence if necessary, the P (m)Nconverge to a nonzero polynomialP N . By continuity, P N (z, j k ϕ(z)) = 0 for all z ∈ O(∆ n ). We claimthat the complement of the union of the F N is dense in O k (∆ n ). Indeed,let ϕ(z) be such that there exists a degree N polynomial P satisfying (8.1).Fix z 0 ∈ ∆ n having rational real and imaginary parts. Then the complexnumbers z 0 , ∂z αϕ(z 0), |α| ≤ k, are algebraically dependent. By a Cantorianargument, there exists complex numbers χ α 0 arbitrarily close to ∂α z ϕ(z 0)such that z 0 , χ α 0 are algebraically independent. Let χ(z) be a polynomialwith ∂z α (z 0 ) = χ α 0 − ∂t α ϕ(z 0 ). We can choose χ to be arbitrarily close tozero in the C k (∆ n ) norm. Then the function ϕ(z) + χ(z) is not Nash algebraic.REFERENCES141[Ar1974][BER1999][BER2000]ARNOLD, V.I.: Équations différentielles ordinaires. Champs de vecteurs,groupes à un paramètre, difféomorphismes, flots, systèmes linéaires, stabilitédes positions d’équilibre, théorie des oscillations, équations différentiellessur les variétés. Traduit du russe par Djilali Embarek, Éditions Mir, Moscou,1974. 267pp.BAOUENDI, M.S.; EBENFELT, P.; ROTHSCHILD, L.P.: Rational dependenceof smooth and analytic CR mappings on their jets. Math. Ann. 315 (1999),205–249.BAOUENDI, M.S.; EBENFELT, P.; ROTHSCHILD, L.P.: Local geometric propertiesof real submanifolds in complex space, Bull. Amer. Math. Soc. 37(2000), no.3, 309–336.

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144Symmetries of partial differential equationsJoël Merker (with Hervé Gaussier)Abstract. We establish a link between the study of completely integrable systemsof partial differential equations and the study of generic submanifolds in C n . Usingthe recent developments of Cauchy-Riemann geometry we provide the set ofsymmetries of such a system with a Lie group structure. Finally we determine theprecise upper bound of the dimension of this Lie group for some specific systemsof partial differential equations.Table of contents1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.2. Submanifold of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.3. Lie theory for partial differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159.4. Optimal upper bound on dim K Sym(E ) when n = m = 1 . . . . . . . . . . . . . . . . . . . .166.5. Optimal upper bound on dim K Sym(E ) in the general dimensional case. . . . . . .170.J. Korean Math. Soc. 40 (2003), no. 3, 517–5611. INTRODUCTIONTo study the geometry of a real analytic Levi nondegenerate hypersurfaceM in C 2 , one of the principal ideas of H. Poincaré, of B. Segre andof É. Cartan in the fundamental memoirs [20], [Se1931], [Se1932], [3] wasto associate to M a system (E M ) of (partial) differential equations, in orderto solve the so-called equivalence problem. Establishing a natural correspondencebetween the local holomorphic automorphisms of M and the Liesymmetries of (E M ) they could use the classification results on differentialequations achieved by S. Lie in [5] and pursued by A. Tresse in [Tr1896].Starting with such a correspondence, we shall establish a general link betweenthe study of a real analytic generic submanifold of codimension min C n+m and the study of completely integrable systems of analytic partialdifferential equations. We shall observe that the recent theories in Cauchy-Riemann (CR) geometry may be transposed to the setting of partial differentialequations, providing some new information on their Lie symmetries.Indeed consider for K = R or C a K-analytic system (E ) of the followinggeneral form:()(E ) u j x α(x) = F αj x, u(x), (u j(q) (x))x β(q) 1≤q≤p .

Here x = (x 1 , . . ., x n ) ∈ K n , u = (u 1 , . . ., u m ) ∈ K m , the integersj(1), . . ., j(p) satisfy 1 ≤ j(q) ≤ m for q = 1, . . ., p, and α and the multiindicesβ(1), . . ., β(p) ∈ N n satisfy |α|, |β(q)| ≥ 1. We also require(j, α) ≠ (j(1), β(1)), . . ., (j(p), β(p)). For j = 1, . . ., m and α ∈ N n , wedenote by u j x the partial derivative α ∂|α| u j /∂x α . We assume that the system(E ) is completely integrable, namely that the Pfaffian system naturallyassociated in the jet space is involutive in the sense of Frobenius. We notethat in that case (E ) is locally solvable, meaning that through every point(x ∗ , u ∗ , u ∗ β , u∗ α) in the jet space, satisfying u ∗ α = F α (x ∗ , u ∗ , u ∗ β ) (written ina condensed form), there exists a local K-analytic solution u = u(x) of(E ) satisfying u(x ∗ ) = u ∗ and u x β(x ∗ ) = u ∗ β . Consequently the Lie theory([Ol1986]) may be applied to such systems. We shall associate with (E ) thesubmanifold of solutions M in K n+2m+p given by K-analytic equations ofthe form(7.28) u j = Ω j (x, ν, χ), j = 1, . . ., m,where ν ∈ K m and where χ ∈ K p . Moreover the integer m + p is the numberof initial conditions for the general solution u(x) := Ω(x, ν, χ) of (E ),whose existence and uniqueness follow from complete integrability. Precisely,the parameters ν, χ correspond to the data u(0), (u j(q)x β(q) (0)) 1≤q≤p . Inthe special case where the system (E ) is constructed from a generic submanifoldM as in [Se1931], [24] (see also Subsection 2.2 below), the correspondingsubmanifold of solutions is exactly the extrinsic complexificationof M.A pointwise K-analytic transformation (x ′ , u ′ ) = Φ(x, u) defined in aneighbourhood of the origin and sufficiently close to the iedntity mappingis called a Lie symmetry of (E ) if it transforms the graph of every∑solution to the graph of an other local solution. A vector field X =nl=1 Ql (x, u) ∂/∂x l + ∑ mj=1 Rj (x, u) ∂/∂u j is called an infinitesimal symmetryof (E ) if for every s close to zero in K the local diffeomorphism(x, u) ↦→ exp(sX)(x, u) associated to the flow of X is a Lie symmetry ofE . According to [Ol1986] (Chapter 2) the infinitesimal symmetries of (E )form a Lie algebra of vector fields defined in a neighbourhood of the originin K n × K m , denoted by Sym(E ). Inspired by recent developments in CRgeometry we shall provide in Section 2 nondegeneracy conditions on M insuringfirstly that Sym(E ) may be identified with the Lie algebra Sym(M )of vector fields of the form(7.28)n∑l=1Q l (x, u) ∂∂x l+m∑j=1R j (x, u) ∂∂u j + m∑j=1Π j (ν, χ) ∂∂ν j + p∑q=1Λ q (ν, χ)145∂∂χ q,

146which are tangent to M , and secondly that Sym(M ) ∼ = Sym(E ) is finitedimensional. The strength of this identification is to provide some (non optimal)bound on the dimension of Sym(E ) for arbitrary systems of partial differentialequations with an arbitrary number of variables, see Theorem 6.4.In the second part of the paper (Sections 3, 4 and 5), using the classical Lietheory (cf. [5], [Ol1986], [Ol1995] and [BK1989]), we provide an optimalupper bound on the dimension of Sym(E ) for a completely integrable K-analytic system (E ) of the following form:(E ) u j x = F j α α (x, u(x), (u x β(x)) 1≤|β|≤κ−1), α ∈ N n , |α| = κ, j = 1, . . .,m.This system is a special case of the system studied in Section 2. Forinstance the homogeneous system (E 0 ) : u j x k1···x kκ(x) = 0 is completelyintegrable. The solutions of (E 0 ) are the polynomials of the formu j (x) = ∑ β∈N n , |β|≤κ−1 λj β xβ , j = 1, . . ., m, where λ j β∈ K and a Lie symmetryof (E 0 ) is a transformation stabilizing the graphs of polynomials ofdegree ≤ κ − 1. We prove the following Theorem:Theorem 6.4. Let (E ) be the K-analytic system of partial differential equationsof order κ ≥ 2, with n independent variables and m dependent variables,defined just above. Assume that (E ) is completely integrable. Thenthe Lie algebra Sym(E ) of its infinitesimal symmetries satisfies the followingestimates:(7.28) {dim K (Sym(E )) ≤ (n + m + 2)(n + m), if κ = 2,dim K (Sym(E )) ≤ n 2 + 2n + m 2 + m Cn+κ−1, if κ ≥ 3,where we denote C κ−1. Moreover the inequalities (7.28) becomeequalities for the homogeneous system (E 0n! (κ−1)!).n+κ−1 := (n+κ−1)!We remark that there is no combinatorial formula interpolating these twoestimates. Theorem 6.4 is a generalization of the following results. Forn = m = 1, S. Lie proved that the dimension of the Lie algebra Sym(E )is less than or equal to 8 if κ = 2 and is less than or equal to κ + 4 ifκ ≥ 3, these bounds being reached for the homogeneous system (cf. [5]).For n = 1, m ≥ 1 and κ = 2, F. González-Gascón and A. González-López proved in [11] that the dimension of Sym (E ) is less than or equalto (m + 3)(m + 1). For n = 1, m ≥ 1 and κ = 2, using the equivalencemethod due to É. Cartan, M. Fels [Fe1995] proved that the dimension ofSym(E ) is less than or equal to m 2 + 4m + 3, with equality if and only ifthe system (E ) is equivalent to the system u j x 2 = 0, j = 1, . . .,m. He alsogeneralized this result to the case n = 1, m ≥ 1, κ = 3. For n ≥ 1, m ≥ 1and κ = 2, A. Sukhov proved in [24] that the dimension of Sym(E ) is lessthan or equal to (n + m + 2)(n + m) (the first inequality in Theorem 6.4),with equality for the homogeneous system u j x k1 x k2= 0.

Consequently, for the case κ = 2, we will only give the general form ofthe Lie symmetries of the homogeneous system (E 0 ) (see Subsection 5.2).We will prove Theorem 6.4 for the case κ ≥ 3. The formulas obtained inSections 3, 4 and 5 were checked with the help of MAPLE release 6.Acknowledgment. This article was written while the first author had a sixmonths delegation position at the CNRS. He thanks this institution for providinghim this research opportunity. The authors are indebted to GérardHenry, the computer ingénieur (LATP, UMR 6632 CNRS), for his technicalsupport.2. SUBMANIFOLD OF SOLUTIONS2.1. Preliminary. Let K = R or C. Let n ≥ 1 and let x = (x 1 , . . ., x n ) ∈N. We denote by K{x} the local ring of K-analytic functions ϕ = ϕ(x)defined in some neighbourhood of the origin in K n . If ϕ ∈ K{x} we denoteby ¯ϕ the function in K{x} satisfying ϕ(x) ≡ ¯ϕ(¯x). Recall that a K-analyticfunction ϕ defined in a domain U ⊂ K n is called K-algebraic (in the senseof Nash) if there exists a nonzero polynomial P = P(X 1 , . . .,X n , Φ) ∈K[X 1 , . . .,X n , Φ] such that P(x, ϕ(x)) ≡ 0 on U. All the considerations inthis paper will be local: functions, submanifolds and mappings will alwaysbe defined in a small connected neighbourhood of some point (most oftenthe origin) in K n .2.2. System of partial differential equations associated to a generic submanifoldof C n+m . Let M be a real algebraic or analytic local submanifoldof codimension m in C n+m , passing through the origin. We assume that Mis generic, namely T 0 M + iT 0 M = T 0 C n+m . Classically (cf. [BER1999])there exists a choice of complex linear coordinates t = (z, w) ∈ C n × C mcentered at the origin such that T 0 M = {Im w = 0} and such that thereexist m complex algebraic or analytic defining equations representing M asthe set of (z, w) in a neighbourhood of the origin in C n+m which satisfy(7.28) w 1 = Θ 1 (z, ¯z, ¯w), . . ....,w m = Θ m (z, ¯z, ¯w).Furthermore, the mapping Θ = (Θ 1 , . . .,Θ m ) satisfies the functional equation(7.28) w ≡ Θ(z, ¯z, Θ(¯z, z, w)),which reflects the reality of the generic submanifold M. It follows inparticular from (7.28) that the local holomorphic mapping C m ∋ ¯w ↦→(Θ j (0, 0, ¯w)) 1≤j≤m ∈ C m is of rank m at ¯w = 0.Generalizing an idea due to B. Segre in [Se1931] and [Se1932], exploitedby É. Cartan in [3] and more recently by A. Sukhov in [24], [25], [Su2002],we shall associate to M a system of partial differential equations. For this,we need some general nondegeneracy condition, which generalizes Levi147

148nondegeneracy. Let l 0 ∈ N with l 0 ≥ 1. We shall assume that M is l 0 -finitely nondegenerate at the origin, cf. [BER1999], [Me2003], [8]. Thismeans that there exist multiindices β(1), . . ., β(n) ∈ N n with |β(k)| ≥ 1for k = 1, . . .,n and max 1≤k≤n |β(k)| = l 0 , and integers j(1), . . .,j(n)with 1 ≤ j(k) ≤ m for k = 1, . . .,n such that the local holomorphic mapping(7.28) (C n+m ∋ (¯z, ¯w) ↦−→ (Θ j (0, ¯z, ¯w)) 1≤j≤m , ( Θ j(k),z β(k)(0, ¯z, ¯w) ) )∈ C m+n1≤k≤nis of rank equal to n + m at (¯z, ¯w) = (0, 0). Here, we denote the partialderivative ∂ |β| Θ j (0, ¯z, ¯w)/∂z β simply by Θ j,z β(0, ¯z, ¯w). Then M is Levinondegenerate at the origin if and only if l 0 = 1. By complexifying thevariables ¯z and ¯w, we get new independent variables ζ ∈ C n and ξ ∈ C mtogether with a complex algebraic or analytic m-codimensional submanifoldM in C 2(n+m) of equations(7.28) w j = Θ j (z, ζ, ξ), j = 1, . . ., m,called the extrinsic complexification of M. In the defining equations (7.28)of M , following [Se1931] and [24], we may consider the “dependentvariables” w 1 , . . .,w m as algebraic or analytic functions of the “independentvariables” z = (z 1 , . . ., z n ), with additional dependence on the extra“parameters” (ζ, ξ) ∈ C n+m . Then by applying the differential operator∂ |α| /∂z α to (7.28), we obtain w j,z α(z) = Θ j,z α(z, ζ, ξ). Writing these equationsfor (j, α) = (j(k), β(k)) with k = 1, . . .,n, we obtain a system ofm + n equations{w j (z) = Θ j (z, ζ, ξ), j = 1, . . ., m,(7.28)w j(k),z β(k)(z) = Θ j(k),z β(k)(z, ζ, ξ), k = 1, . . ., n.In this system (7.28), by the assumption of l 0 -finite nondegeneracy (7.28),the algebraic or analytic implicit function theorem allows to solve the parameters(ζ, ξ) in terms of the variables (z k , w j (z), w j(k),z β(k)(z)), providinga local algebraic or analytic C n+m -valued mapping R such that (ζ, ξ) =R ( z k , w j (z), w j(k),z β(k)(z) ) . Finally, for every pair (j, α) different from(1, 0), . . ., (m, 0), (j(1), β(1)), . . ., (j(n), β(n)), we may replace (ζ, ξ) byR in the differentiated expression w j,z α(z) = Θ j,z α(z, ζ, ξ). This yields(w j,z α(z) = Θ j,z α z, R(zk , w j (z), w j(k),z β(k)(z)) )(7.28)(=: F j,α zk , w j (z), w j(k),z β(k)(z) ) .This is the system of partial differential equations associated with M . Asargued by B. Segre in [Se1931], the geometric study of generic submanifoldsof C n may gain much information from the study of their associatedsystems of partial differential equations (cf. [24], [25]). The next paragraphs

are devoted to provide a general one-to-one correspondence between completelyintegrable systems of analytic partial differential equations and theirassociated “submanifolds of solutions” (to be defined precisely below) likeM above. Afterwards, we shall observe that conversely, the study of systemsof analytic partial differential equations also gains much informationfrom the direct study of their associated submanifolds of solutions.1492.3. Completely integrable systems of partial differential equations.Let now n, m, p ∈ N with n, m, p ≥ 1, let κ ∈ N with κ ≥ 2and let u = (u 1 , . . .,u m ) ∈ K m . Consider a collection of p multiindicesβ(1), . . ., β(p) ∈ N n with |β(q)| ≥ 1 for q = 1, . . .,p andmax 1≤q≤p |β(q)| = κ − 1. Consider also p integers j(1), . . ., j(p) with1 ≤ j(q) ≤ m for q = 1, . . ., p. Inspired by (7.28), we consider ageneral system of partial differential equations of n independent variables(x 1 , . . ., x n ) and m dependent variables (u 1 , . . .,u m ) which is of the followingform:()(E ) u j x α(x) = F αj x, u(x), (u j(q) (x))x β(q) 1≤q≤p ,where (j, α) ≠ (j(1), β(1)), . . ., (j(p), β(p)) and j = 1, . . .,m, |α| ≤ κ.Here, we assume that u = 0 is a local solution of the system (E ) and thatthe functions Fα j are K-algebraic or K-analytic in a neighbourhood of theorigin in K n+m+p . Among such systems are included ordinary differentialequations of any order κ ≥ 2, systems of second order partial differentialequation as studied in [24], etc.Throughout this article, we shall assume the system (E ) completely integrable.By analyzing the application of the Frobenius theorem in jet spaces,one can show (we will not develop this) that the general solution of the system(E ) is given by u(x) := Ω(x, ν, χ), where the parameters ν ∈ K nand χ ∈ K n essentially correspond to the “initial conditions” u(0) and(u j(q) (0))x β(q) 1≤q≤p , and Ω is a K-analytic K n -valued mapping. In the case ofa generic submanifold as in Subsection 2.2 above, we recover the mappingΘ. In the sequel, we shall use the following terminology: the coordinates(x, u) will be called the variables and the coordinates (ν, χ) will be calledthe parameters or the initial conditions. In Subsection 2.5 below, we shallintroduce a certain duality where the rôles between variables and parametersare exchanged.2.4. Associated submanifold of solutions. The existence of the functionΩ and the analogy with Subsection 2.2 leads us to introduce the submanifoldof solutions associated to the completely integrable system (E ), whichby definition is the m-codimensional K-analytic submanifold of K n+2m+p ,

150equipped with the coordinates (x, u, ν, χ), defined by the Cartesian equations(7.28) u j = Ω j (x, ν, χ), j = 1, . . .,m.Let us denote this submanifold by M . We stress that in general such a submanifoldcannot coincide with the complexification of a generic submanifoldof C m+n , for instance because K may be equal to R or, if K = C,because the integer p is not necessarily equal to n. Also, even if K = C andn = p, the mapping Ω does not satisfy a functional equation like (7.28). Infact, it may be easily established that the submanifold of solutions of a completelyintegrable system of partial differential equations like (E ) coincideswith the complexification of a generic submanifold if and only if K = C,p = n and the mapping Ω satisfies a functional equation like (7.28).Let now M be a submanifold of K n+2n+p of the form (7.28), but notnecessarily constructed as the submanifold of solutions of a system (E ). Weshall always assume that Ω j (0, ν, χ) ≡ ν j . We say that M is solvable withrespect to the parameters if there exist multiindices β(1), . . ., β(p) ∈ N nwith |β(q)| ≥ 1 for q = 1, . . .,p and integers j(1), . . ., j(p) with 1 ≤j(q) ≤ m for q = 1, . . ., p such that the local K-analytic mapping(7.28) (K m+p ∋ (ν, χ) ↦−→ (Ω j (0, ν, χ) 1≤j≤m , ( Ω j(q),x β(q)(0, ν, χ) ) )∈ K m+p1≤q≤pis of rank equal to m+p at (ζ, χ) = (0, 0) (notice that since Ω j (0, ν, χ) ≡ ν j ,then the m first components of the mapping (7.28) are already of rank m).We remark that the submanifold of solutions of a system (E ) is automaticallysolvable with respect to the variables, the multiindices β(q) and theintegers j(q) being the same as in the arguments of the right hand side termsF j α in (E ).2.5. Dual system of defining equations. Since Ω j (0, ν, χ) ≡ ν j , we maysolve the equations (7.28) with respect to ν by means of the analytic implicitfunction theorem, getting an equivalent system of equations for M :(7.28) ν j = Ω ∗ j(χ, x, u), j = 1, . . ., m.We call this the dual system of defining equations for M . By construction,we have the functional equation(7.28) u ≡ Ω(x, Ω ∗ (χ, x, u), χ),implying the identity Ω ∗ j (0, x, u) ≡ uj . We say that M is solvable withrespect to the variables if there exist multiindices δ(1), . . ., δ(n) ∈ N p with|δ(l)| ≥ 1 for l = 1, . . .,n and integers j(1), . . ., j(n) with 1 ≤ j(l) ≤ m

for l = 1, . . .,m such that the local K-analytic mapping(7.28) (() )K n+m ∋ (x, u) ↦−→ (Ω ∗ j(0, x, u)) 1≤j≤m , Ω ∗ j(l), χ(0, x, u) ∈ K m+nδ(l)1≤l≤nis of rank equal to n+m at (x, u) = (0, 0) (notice that since Ω ∗ j(0, x, u) ≡ u j ,the m fisrt components of the mapping (7.28) are already of rank m).In the case where M is the complexification of a generic submanifoldthen the solvability with respect to the parameters is equivalent to the solvabilitywith respect to the variables since Ω ∗ ≡ Ω. However we notice thata submanifold M of solutions of a system (E ) is not automatically solvablewith respect to the variables, as shows the following trivial example.Example 1. Let n = 2, m = 1 and let (E ) denote the system u x2 = 0,u x1 x 1= 0, whose general solutions are u(x) = ν + x 1 χ =: Ω(x 1 , x 2 , ν, χ).Notice that the variable x 2 is absent from the dual equation ν = u −x 1 χ 1 =:Ω ∗ (χ, x 1 , x 2 , u). It follows that M is not solvable with respect to the variables.2.6. Symmetries of (E ), their lift to the jet space and their lift to theparameter space. We denote by Jn,m κ the space of jets of order κ of K-analytic mappings u = u(x) from K n to K m . Let(7.28) (x l , u j , U i 1l 1, U i 1l 1 ,l 2, . . .,U i 1l 1 ,...,l κ) ∈ K n+m Cκ κ+ndenote the natural coordinates on Jn,m κ . Here, the superscripts j, i 1 andthe subscripts l, l 1 , l 2 , . . .,l κ satisfy j, i 1 = 1, . . .,m and l, l 1 , l 2 , . . .,l κ =1, . . ., n. The independent coordinate U i 1l 1 ,...,l λcorresponds to the partial derivativeu i 1xl1 ...x lλ. Finally, by symmetry of partial differentiation, we identityevery coordinate U i 1l 1 ,...,l λwith the coordinates U i 1σ(l 1 ),...,σ(l λ ), where σ is an arbitrarypermutation of the set {1, . . ., λ}. With these identifications, the κ-thorder jet space Jn,m κ is of dimension n + m Cκ+n, κ where Cp q := p! denotesthe binomial coefficient. Also, we shall sometimes use an equivalentq! (p−q)!notation for coordinates on Jn,m:κ(7.28) (x l , u j , U i β ) ∈ Kn+m Cn κ+n ,where β ∈ N n satisfies |β| ≤ κ and where the independent coordinate Uβicorresponds to the partial derivative u i x. βassociated to the system (E ) is the so-called skeleton ∆ E , which is theK-analytic submanifold of dimension n+m+p in Jn,m κ simply defined byreplacing the partial derivatives of the dependent variables u j by the independentjet variables in (E ):()(7.28) Uα j = Fαj x, u, (U j(q)β(q) ) 1≤q≤p ,151

152for (j, α) ≠ (j(1), β(1)), . . ., (j(p), β(p)) and j = 1, . . .,m, |α| ≤ κ.Clearly, the natural coordinates on the submanifold ∆ E of Jn,m κ are then + m + p coordinates()(7.28)x, u, (U j(q)β(q) ) 1≤q≤p .Let h = h(x, u) be a local K-analytic diffeomorphism of K n+m close to theidentity mapping and let π κ : J κ n,m → Kn+m be the canonical projection.According to [Ol1986] (Chapter 2) there exists a unique lift h (κ) of h toJ κ n,m such that π κ ◦h (κ) = h◦π κ . The components of h (κ) may be computedby means of universal combinatorial formulas and they are rational functionsof the jet variables (7.28), their coefficients being partial derivatives of thecomponents of h, see for instance §3.3.5 of [BK1989]. By definition, h is alocal symmetry of (E ) if h transforms the graph of every local solution of(E ) into the graph of another local solution of (E ). This definition seems tobe rather uneasy to handle, because of the abstract quantification of “everylocal solution”, but we have the following concrete characterization for h tobe a local symmetry of (E ), cf. Chapter 2 in [Ol1986].Lemma 8.1. The following conditions are equivalent:(1) The local transformation h is a local symmetry of (E ).(2) Its κ-th prolongation h (κ) is a local self-transformation of the skeleton∆ E of (E ).These considerations have an infinitesimal version. Indeed, let X =∑ nl=1 Ql (x, u) ∂/∂x l + ∑ mj=1 Rj (x, u) ∂/∂u j be a local vector field withK-analytic coefficients which is defined in a neighbourhood of the origin inK n+m . Let s ∈ K and consider the flow of L as the one-parameter familyh s (x, u) := exp(s X)(x, u) of local transformations. We recall that X isan infinitesimal symmetry of (E ) if for every small s ∈ K, the mappingh s (x, u) := exp(s X)(x, u) is a local symmetry of (E ). By differentiatingwith respect to s the κ-th prolongation (h s ) (κ) of h s at s = 0, we obtaina unique vector field X (κ) on the κ-th jet space, called the κ-th prolongationof X and which satisfies (π k ) ∗ (X (κ) ) = X. In Subsections 3.1 and 3.2below, we shall analyze the combinatorial formulas for the coefficients ofX (κ) , since they will be needed to prove Theorem 6.4.Let X E be the projection to the restricted jet space K m+n+p , equippedwith the coordinates (7.28), of the restriction of X (κ) to ∆ E , namely(7.28) X E := (π κ,p ) ∗ (X (κ) | ∆E ).The following Lemma, called the Lie criterion, is the concrete characterizationfor X to be an infinitesimal symmetry of (E ) and is a direct corollaryof Lemma 8.1, cf. Chapter 2 in [Ol1986]. This criterion will be central inthe next Sections 3, 4 and 5.

Lemma 8.1. The following conditions are equivalent:(1) The vector field X is an infinitesimal symmetry of (E ).(2) Its κ-th prolongation X (κ) is tangent to the skeleton ∆ E .We denote by Sym(E ) the set of infinitesimal symmetries of (E ). Sinceit may be easily checked that (cX + dY ) (κ) = cX (κ) + dY (κ) and that[X (κ) , Y (κ) ] = ([X, Y ]) (κ) , see Theorem 2.39 in [Ol1986], it follows fromLemma 8.1 (2) that Sym(E ) is a Lie algebra of locally defined vector fields.Our main question in this section is the following: under which naturalconditions is Sym(E ) finite-dimensional ?Example 2. We observe that the Lie algebra Sym(E ) of the system (E )presented in Example 1 is infinite-dimensional, since it includes all vectorfields of the form X = Q 2 (x 1 , x 2 , u) ∂/∂x 2 , as may be verified. As we willargue in Proposition 3.1 below, this phenomenon is typical, the main reasonlying in the first order relation u x2 = 0.By analyzing the construction of the submanifold of solutions M associatedto the system (E ), we may establish the following correspondence (weshall not develop its proof).Proposition∑3.1. To every infinitesimal symmetry X =nl=1 Ql (x, u) ∂/∂x l + ∑ mj=1 Rj (x, u) ∂/∂u j of (E ), there corresponds aunique vector field of the form(7.28) X =m∑j=1Π j (ν, χ) ∂∂ν j + p∑q=1Λ q (ν, χ) ∂∂χ q,whose coefficients depend only on the parameters (ν, χ), such that X + Xis tangent to the submanifold of solutions M .This leads us to define the Lie algebra Sym(M ) of vector fields of theform(7.28)n∑l=1Q l (x, u) ∂∂x l+m∑j=1R j (x, u) ∂∂u j + m∑j=1Π j (ν, χ) ∂∂ν j + p∑q=1153Λ q ∂(ν, χ)∂χ qwhich are tangent to M . We shall say that the submanifold M is degenerateif there exists a nonzero vector field of the form X = ∑ n∑ l=1 Ql (x, u) ∂/∂x l +mj=1 Rj (x, u) ∂/∂u j which is tangent to M , which means that the correspondingX part is zero. In this case, we claim that Sym(M ) isinfinite dimensional. Indeed there exists then a nonzero vector fieldT = ∑ nl=1 Ql (x, u)∂/∂x l + ∑ mj=1 Rj (x, u)∂/∂u j tangent to M . Consequently,for every K-analytic function A(x, u), the vector field A(x, u) Tbelongs to Sym(M ), hence Sym(M ) is infinite dimensional.

154By developing the dual defining functions of M with respect to the powersof χ, we may write(7.28) ν j = Ω ∗ j(χ, x, u) = ∑ γ∈N p χ γ Ω ∗ j,γ(x, u),where the functions Ω ∗ j,γ(x, u) are K-analytic in a neighbourhood of the origin,we may formulate a criterion for M to be non degenerate with respectto the variables (whose proof is skipped).Proposition 3.1. The submanifold M is not degenerate with respect to thevariables if and only if there exists an integer k such that the generic rank ofthe local K-analytic mapping(7.28) (x, u) ↦−→ ( Ω ∗ j,γ (x, u)) 1≤j≤m, γ∈N p , |γ|≤kis equal to n + m.Seeking for conditions which insure that Sym(M ) is finite-dimensional,it is therefore natural to assume that the generic rank of the mapping (7.28)is equal to n + m. Furthermore, to simplify the presentation, we shall assumethat the rank at (x, u) = (0, 0) (not only the generic rank) of themapping (7.28) is equal to n + m for k large enough. This is a “Zariskigeneric”assumption. Coming back to (7.28), we observe that this meansexactly that M is solvable with respect to the variables. Then we denote byl ∗ 0 the smallest integer k such that the rank at (x, u) = (0, 0) of the mapping(7.28) is equal to n + m and we say that M is l ∗ 0-solvable with respectto the variables. Also, we denote by l 0 the integer max 1≤q≤p |β(q)| and wesay that M is l 0 -solvable with respect to the parameters.2.7. Fundamental isomorphism between Sym(E ) and Sym(M ). In theremainder of this Section 2, we shall assume that M is l 0 -solvable withrespect to the parameters and l ∗ 0-solvable with respect to the variabes. Inthis case, viewing the variables (ν 1 , . . .,ν m ) in the dual equations ν j =Ω ∗ j(χ, x, u) of M as a mapping of χ with (dual) “parameters” (x, u) andproceeding as in Subsection 2.2, we may construct a dual system of completelyintegrable partial differential equations of the form()(E ∗ ) ν j χ γ(χ) = Gj γ χ, ν(χ), (ν j(l) (χ))χ δ(l) 1≤l≤n ,where (j, γ) ≠ (j(1), δ(1)), . . ., (j(n), δ(n)). This system has its own infinitesimalsymmetry Lie algebra Sym(E ∗ ).Theorem 6.4. If M is both solvable with respect to the parameters andsolvable with respect to the variables, we have the following two isomorphisms:(7.28) Sym(E ) ∼ = Sym(M ) ∼ = Sym(E ∗ ),

155namely X ←→ X + X ←→ X .In Subsection 2.10 below, we shall introduce a second geometric conditionwhich is in general necessary for Sym(M ) to be finite-dimensional.2.8. Local (pseudo)group Sym(M ) of point transformations of M . Weshall study the geometry of a local K-analytic submanifold M of K n+2m+pwhose equations and dual equations are of the form{u j = Ω j (x, ν, χ), j = 1, . . .,m,(7.28)ν j = Ω ∗ j (χ, x, u), j = 1, . . ., m.Let t := (x, u) ∈ K n+m and τ := (ν, χ) ∈ K n+m . We are interested indescribing the set of local K-analytic transformations of the space K n+2m+pwhich are of the specific form(7.28) (t, τ) ↦−→ (h(t), φ(τ)),and which stabilize M , in a neighborhood of the origin. We denote the localLie pseudogroup of such transformations (possibly infinite-dimensional) bySym(M ). Importantly, each transformation of Sym(M ) stabilize both thesets {t = ct.} and the sets {τ = ct.}. Of course, the Lie algebra of Sym(M )coincides with Sym(M ) defined above.2.9. Fundamental pair of foliations on M . Let p 0 ∈ K n+2m+p be a fixedpoint of coordinates (t p0 , τ p0 ). Firstly, we observe that the intersection M ∩{τ = τ p0 } consists of the n-dimensional K-analytic submanifold of equationu = Ω(x, τ p0 ). As τ p0 varies, we obtain a local K-analytic foliation of M byn-dimensional submanifolds. Let us denote this first foliation by F p and callit the foliation of M with respect to parameters. Secondly, and dually, weobserve that the intersection M ∩{t = t p0 } consists of the p-dimensional K-analytic submanifold of equation ν = Ω ∗ (χ, t p0 ). As t p0 varies, we obtaina local K-analytic foliation of M by p-dimensional submanifolds. Let usdenote this second foliation by F v and call it the foliation of M with respectto the variables. We call (F p , F v ) the fundamental pair of foliations on M .2.10. Covering property of the fundamental pair of foliations. We wishto formulate a geometric condition which says that starting from the originin M and following alternately the leaves of F p and the leaves of F v , wecover a neighborhood of the origin in M . Let us introduce two collections

156(L k ) 1≤k≤n and (Lq ∗ ) 1≤q≤p of vector fields whose integral manifolds coincidewith the leaves of F p and F v :(7.28)⎧⎪⎨⎪⎩L k := ∂∂x k+L ∗q := ∂∂χ q+m∑j=1m∑j=1∂Ω j(x, ν, χ) ∂∂x k ∂u , j∂Ω ∗ j∂χ q(χ, x, u) ∂∂ν j ,k = 1, . . .,n,k = 1, . . .,n.Let p 0 be a fixed point in M of coordinates (x p0 , u p0 , ν p0 , χ p0 ) ∈ K n+2m+p ,let x 1 := (x 1,1 , . . .,x 1,n ) ∈ K n be a “multitime” parameter and define themultiple flow map(7.28){Lx1 (x p0 , u p0 , ν p0 , χ p0 ) := exp(x 1 L )(p 0 ) := exp(x 1,n L n (· · ·(exp(x 1,1 L 1 (p 0 ))) · · ·)) :=:= (x p0 + x 1 , Ω(x p0 + x 1 , ν p0 , χ p0 ), ν p0 , χ p0 ).Similarly, for χ = (χ 1,1 , . . .,χ 1,p ) ∈ K p , define the multiple flow map(7.28)L ∗ χ 1(x p0 , u p0 , ν p0 , χ p0 ) := (x p0 , u p0 , Ω ∗ (χ p0 + χ 1 , x p0 , u p0 ), χ p0 + χ 1 ).We may define now the mappings which correspond to start from the originand to move alternately along the two foliations F p and F v . If the firstmovement consists in moving along the foliation F v , we define⎧Γ 1 (x 1 ) := L x1 (0),⎪⎨ Γ 1 (x 1 , χ 1 ) := Lχ ∗(7.28)1(L x1 (0)),Γ 3 (x 1 , χ 1 , x 2 ) := L x2 (Lχ ⎪⎩∗ 1(L x1 (0))),Γ 4 (x 1 , χ 1 , x 2 , χ 2 ) := Lχ ∗ 2(L x2 (Lχ ∗ 1(L x1 (0)))).Generally, we may define the maps Γ k ([xχ] k ), where [xχ] k =(x 1 , χ 1 , x 2 , χ 2 , . . .) with exactly k terms and where each x l belongsto K n and each χ l belongs to K p . On the other hand, if the first movementconsists in moving along the foliation F p , we start with Γ ∗ 1 (χ 1) := Lχ ∗ 1(0),Γ ∗ 2(χ 1 , x 1 ) := L x1 (Lχ ∗ 1(0)), etc., and generally we may define the mapsΓ ∗ k ([χx] k), where [χx] k = (χ 1 , x 1 , χ 2 , x 2 , . . .), with exactly k terms. Therange of both maps Γ k and Γ ∗ k is contained in M . We call Γ k the k-th chainand Γ ∗ k the k-th dual chain.Definition 5.3. The pair of foliations (F p , F v ) is called covering at the originif there exists an integer k such that the generic rank of Γ k is (maximalpossible) equal to dim K M . Since the dual (k +1)-th chain Γ ∗ k+1 for χ 1 = 0identifies with the k-th chain Γ k , it follows that the same property holds forthe dual chains.

In terms of Sussmann’s approach [27], this means that the local orbit ofthe two systems of vector fields (L k ) 1≤k≤n and (Lq ∗)1≤q≤p is of maximal dimension.Reasoning as in [27] (using the so-called backward trick in ControlTheory, see also [Me2003]), it may be shown that there exists the smallesteven integer 2µ 0 such that the ranks of the two maps Γ 2µ0 and Γ ∗ 2µ 0at theorigin (not only their generic rank) in K nµ 0+pµ 0are both equal to dim K M .This means that Γ 2µ0 and Γ ∗ 2µ 0are submersive onto a neighborhood of theorigin in M . We call µ 0 the type of the pair of foliations (F p , F v ). It mayalso be established that µ 0 ≤ m + 2.Example 2.46. We give an example of a submanifold which is both 1-solvable with respect to the parameters and with respect to the variablesbut whose pair of foliations is not covering: with n = 1, m = 2 andp = 1, this is given by the two equations u 1 = ν 1 , u 2 = ν 2 + xχ 1 .Then Sym(M ) is infinite-dimensional since it contains the vector fieldsa(u 1 ) ∂/∂u 1 +a(ν 1 ) ∂/∂ν 1 , where a is an arbitrary K-analytic function. Forthis reason, we shall assume in the sequel that the pair of foliations (F p , F v )is covering at the origin.2.11. Estimate on the dimension of the local symmetry group of thesubmanifold of solutions. We may now formulate the main theorem ofthis section, which shows that, under suitable nondegeneracy conditions,Sym(M ) is a finite dimensional local Lie group of local transformations. Ift ∈ K n+m , we denote by |t| := max 1≤k≤n+m |t k |. If (h, φ) ∈ Sym(M )we denote by J k t h(0) the k-th order jet of h at the origin and by J k τ φ(0) thek-th order jet of φ at the origin. Also, we shall assume that M is eitherK-algebraic or K-analytic. Of course, the K-algebraicity of the submanifoldof solutions does not follow from the K-algebraicity of the right hand sidesF j α of the system of partial differential equations (E ).Theorem 6.4. Assume that the K-algebraic or K-analytic submanifold ofsolutions M of the completely integrable system of partial differential equations(E ) is both l 0 -sovable with respect to the parameters and l ∗ 0 -solvablewith respect to the variables. Assume that the fundamental pair of foliations(F p , F v ) is covering at the origin and let µ 0 be its type at the origin. Thenthere exists ε 0 > 0 such that for every ε with 0 < ε < ε 0 , the following fourproperties hold:(a) The (pseudo)group Sym(M ) of local K-analytic diffeomorphisms definedfor {(t, τ) ∈ K n+2m+p : |t| < ε, |τ| < ε} which are of theform (t, τ) ↦→ (h(t), φ(τ)) and which stabilize M is a local Lie pseudogroupof transformations of finite dimension d ∈ N.(b) Let κ 0 := µ 0 (l 0 + l ∗ 0). Then there exist two K-algebraic or K-analyticmappings H κ0 and Φ κ0 which depend only on M and which may beconstructed algorithmically by means of the defining equations of M157

158such that every element (h, φ) ∈ Sym(M ), sufficiently close to theidentity mapping, may be represented by{ h(t) = Hκ0 (t, J κ 0t h(0)),(7.28)φ(τ) = Φ κ0 (τ, J κ 0τ φ(0)).Consequently, every element of Sym(M ) is uniquely determined byits κ 0 -th jet at the origin and the dimension d of the Lie algebraSym(M ) is bounded by the number of components of the vector(J κ 0t h(0), J κ 0τ φ(0)), namely we have(7.28)dim K Sym(E ) = dim K Sym(M ) ≤ (n+m) C κ 0n+m+κ 0+(m+p) C κ 0m+p+κ 0.(c) There exists ε ′ with 0 < ε ′ < ε and a K-algebraic or K-analytic mapping(H M , Φ M ) which may be constructed algorithmically by meansof the defining equations of M , defined in a neighbourhood of theorigin in K n+2m+p × K d with values in K n+2m+p and which satifies(H M (t, 0), Φ M (τ, 0)) ≡ (t, τ), such that every element (h, φ) ∈Sym(M ) defined on the set {(t, τ) ∈ K n+2m+p : |t| < ε ′ , |τ| < ε ′ },sufficiently close to the identity mapping and stabilizing M may berepresented as (h(t), φ(τ)) ≡ (H M (t, s h,φ ), Φ M (τ, s h,φ )) for a uniqueelement s h,φ ∈ K d depending on the mapping (h, φ).(d) The mapping (t, τ, s) ↦−→ (H M (t, s), Φ M (τ, s)) defines a local K-algebraic or K-analytic Lie group of local K-algebraic or K-analytictransformations stabilizing M .2.12. Applications. The proof of Theorem 6.4, which possesses strongsimilarities with the proof of Theorem 4.1 in [8], will not be presented.It seems that Theorem 6.4, together with the argumentation on the necessityof assumptions that M be solvable with respect to the variables andthat its fundamental pair of foliations be covering, is a new result aboutthe finite-dimensionality of a completely integrable system of partial differentialequations having an arbitrary number of independent and dependentvariables. The main interest lies in the fact that we obtain the algorithmicallyconstructible representation formula (7.28) together with the local Lie groupstructure mapping (H M , Φ M ). In particular, we get as a corollary that everytransformation (h(t), φ(τ)) given by a formal power series (not necessarilyconvergent) is as smooth as the applications (H κ0 , Φ κ0 ) are, namely everyformal element of Sym(M ) is necessarily K-algebraic or K-analytic. As acounterpart of its generality, Theorem 3 does not provide optimal bounds,as shows the following illustration.Example 2.46. Let n = m = 1, let κ ≥ 3 and let (E ) denote the ordinarydifferential equation u x κ(x) = F(x, u(x), u x (x), . . .,u x κ−1(x)). Thenthe submanifold of solutions M is of the form u = ν + xχ 1 + · · · +

x κ−1 χ κ−1 + O(|x| κ ) + O(|χ| 2 ). It may be checked that l 0 = κ − 1, l ∗ 0 = 1and µ 0 = 3, hence κ 0 = 3κ. Then the dimension estimate in (7.28) is:dim K Sym(E ) ≤ 2 C2+3κ + κ C3κ 4κ . This bound is much larger than the optimalbound dim K Sym(E ) ≤ κ + 4 due to S. Lie (cf. [5]; see also the casen = m = 1 of Theorem 6.4).Untill now we focused on providing the set of Lie symmetries of a generalsystem of partial differential equations with a local Lie group structure. Asa byproduct we obtained the (non optimal) dimensional upper bound (7.28)of Theorem 6.4. In the next Sections 3, 4 and 5, using the classical Liealgorithm based on the Lie criterion (see Lemma 8.1), we provide an optimalbound for some specific systems of partial differential equations, answeringan open problem raised in [Ol1995] page 206.1593. LIE THEORY FOR PARTIAL DIFFERENTIAL EQUATIONS3.1. Prolongation of vector fields to the jet spaces. Consider the followingK-analytic system (E ) of non linear partial differential equations:)(7.28) u j x k1···x kκ(x) = F j k 1 ,...,k κ(x, u(x), u i 1xl1(x), . . .,u i 1xl1···x lκ−1(x) ,where 1 ≤ k 1 ≤ · · · ≤ k κ ≤ n, 1 ≤ j ≤ m, and F j k 1 ,...,k κare analyticfunctions of n + m Cn+κ−1 variables, defined in a neighbourhood of theorigin. We assume that (E ) is completely integrable. The Lie theory consistsin studying the infinitesimal symmetries X = ∑ n∑ l=1 Ql (x, u) ∂/∂x l +mj=1 Rj (x, u) ∂/∂u j of (E ). Consider the skeleton of (E ), namely thecomplex subvariety ∆ E of codimension m Cκ+n−1 κ in the jet space Jn,m,κdefined by)(7.28) U j k 1 ,...,k κ= F j k 1 ,...,k κ(x, u, U i 1l 1, . . ., U i 1l 1 ,...,l κ−1,where j, i 1 = 1, . . ., m and k 1 , . . .,k κ , l 1 , . . .,l κ−1 = 1, . . ., n. For k =1, . . ., n let D k be the k-th operator of total differentiation, characterized bythe property that for every integer λ ≥ 2 and for every analytic functionP = P(x, u, U i 1l 1, . . ., U i 1l 1 ,...,l λ−1) defined in the jet space Jn,m λ−1 , the operatorD k is the unique formal infinite differential operator satisfying the relation(7.28)⎧⎪⎨⎪⎩([D k P]x, u(x), u i 1xl1(x), . . . , u i 1∂[P∂x k)xl1···x lλ−1(x) ≡(x, u(x), u i 1xl1(x), . . .,u i 1xl1···x lλ−1(x))].

160Note that this identity involves only the troncature of D k to order λ, denotedby Dk λ , and defined by⎧Dk ⎪⎨λ := ∂ m∑+ U i ∂1k∂x k ∂u + ∑ m n∑U i ∂1i 1 k,l 1i 1 =1i 1∂U i + · · ·+1=1 l 1 =1 l 1(7.28)m∑ n∑+ U ⎪⎩i ∂1k,l 1 ,...,l λ−1∂U i .1l 1 ,...,l λ−1i 1 =1 l 1 ,...,l λ−1 =1According to Theorem 2.36 of [Ol1986], the prolongation of order κ of avector field X = ∑ nl=1 Ql (x, u) ∂/∂x l + ∑ mj=1 Rj (x, u) ∂/∂u j , denoted byX (κ) , is the unique vector field on the space Jn,m κ of the form(7.28) ⎧X ⎪⎨(κ) = X +⎪⎩m∑j=1+n∑k 1 =1m∑j=1R j k 1∂∂U j +k 1n∑k 1 ,...,k κ=1m∑j=1R j k 1 ,...,k κn∑k 1 ,k 2 =1R j k 1 ,k 2∂∂U j k 1 ,k 2 ,...,k κ,∂∂U j k 1 ,k 2+ · · ·+corresponding to the infinitesimal action of the flow of X on the jets of orderκ of the graphs of maps u = u(x), and whose coefficients are computedrecursively by the formulas⎧n∑R j k 1:= Dk 1 1(R j ) − Dk 1 1(Q l 1) U j l 1,(7.28)⎪⎨R j k 1 ,k 2:= D 2 k 2(R j k 1) −l 1 =1n∑Dk 1 2(Q l 1) U j k 1 ,l 1,l 1 =1· · · · · · · · · · · · · · · · · · · · · · · · · · ·n∑R ⎪⎩j k 1 ,k 2 ,...,k λ:= Dk λ λ(R j k 1 ,...,k λ−1) − Dk 1 λ(Q l 1) U j k 1 ,...,k λ−1 ,l 1.For a better comprehension of the general computation, let us start by computingR κ in the case n = m = 1.3.2. Computation of R κ when n = m = 1. A direct application of thepreceding formulas leads to the following classical expressions:(7.28) ⎧⎪⎨R 1 = R x + [R u − Q x ] U 1 + [−Q u ] (U 1 ) 2 .R 2 = R x 2 + [2R xu − Q x 2] U 1 + [R u 2 − 2Q xu ] (U 1 ) 2 + [−Q u 2] (U 1 ) 3 +⎪⎩+ [R u − 2Q x ] U 2 + [−3Q u ] U 1 U 2 .l 1 =1

Observe that these expressions are polynomial in the jet variables, their coefficientsbeing differential expressions involving a partial derivative of R(with a positive integer coefficient) and a partial derivative of Q (with a negativeinteger coefficient). We have also:(7.28) ⎧R 3 = R x 3 + [3R x 2 u − Q x 3] U 1 + [3R xu 2 − 3Q x 2 u] (U 1 ) 2 +⎪⎨R 4 = R x 4 + [4R x 3 u − Q x 4] U 1 + [6R x 2 u 2 − 4Q x 3 u] (U 1 ) 2 ++ [4R xu 3 − 6Q x 2 u 2] (U1 ) 3 + [R u 4 − 4Q xu 3] (U 1 ) 4 + [−Q u 4] (U 1 ) 5 +⎪⎩161+ [R u 3 − 3Q xu 2] (U 1 ) 3 + [−Q u 3] (U 1 ) 4 + [3R xu − 3Q x 2] U 2 ++ [3R u 2 − 9Q xu ] U 1 U 2 + [−6Q u 2] (U 1 ) 2 U 2 + [−3Q u ] (U 2 ) 2 ++ [R u − 3Q x ] U 3 + [−4Q u ] U 1 U 3 .+ [6R x 2 u − 4Q x 3] U 2 + [12R xu 2 − 18Q x 2 u] U 1 U 2 ++ [6R u 3 − 24Q xu 2] (U 1 ) 2 U 2 + [−10Q u 3] (U 1 ) 3 U 2 ++ [3R u 2 − 12Q xu ] (U 2 ) 2 + [−15Q u 2] U 1 (U 2 ) 2 + [4R xu − 6Q x 2] U 3 ++ [4R u 2 − 16Q xu ] U 1 U 3 + [−10Q u 2] (U 1 ) 2 U 3 + [−10Q u ] U 2 U 3 ++ [R u − 4Q x ] U 4 + [−5Q u ] U 1 U 4 .Remark that all the brackets involved in equations (7.28) are of the form[λ R x a u − µ Q b+1 x a+1 ub], where λ, µ ∈ N and a, b ∈ N.In what follows we will not need the complete form of R κ but only thefollowing partial form:Lemma 8.1. For κ ≥ 4:(7.28)⎧R κ = R x κ + [ Cκ 1 R x κ−1 u − Q x κ]U 1 + [ Cκ 2 R x κ−2 u − Cκ 1 Q ]x κ−1 U 2 +⎪⎨⎪⎩+ [ C 2 κ R x 2 u − C 3 κ Q x 3 ]U κ−2 + [ C 1 κ R xu − C 2 κ Q x 2 ]U κ−1 ++ [ C 1 κ R u 2 − κ2 Q xu]U 1 U κ−1 + [ −C 2 κ+1 Q u]U 2 U κ−1 ++ [ R u − C 1 κ Q x]U κ + [ −C 1 κ+1 Q u]U 1 U κ ++ Remainder,where the term Remainder denotes the remaining terms in the expansion ofR κ .We note that the formula (7.28) is valid for κ = 3, comparing with (7.28),with the convention that the terms U κ−2 and U κ−1 vanish (they coincide withU 1 and U 2 ), and replacing the coefficient −C 2 κ+1 Q u = −C 2 4 Q u = −6 Q uof the monomial U 2 U κ−1 by −3 Q u , as it appears in (7.28). The proof goesby a straightforward computation, applying the recursive definition of thispartial formula.

1623.3. Computation of R κ in the general case. Following the exact samescheme as in the case n = 1 we give the general partial formula forR κ . We start with the first three families of coefficients R j k 1, R j k 1 ,k 2andR j k 1 ,k 2 ,k 3. Let δp q be the Kronecker symbol, equal to 1 if p = q and to 0 ifp ≠ q. More generally, the generalized Kronecker symbols are defined byδ q 1,...,q kp 1 ,...,p k:= δ q 1p 1δ q 2p 2 · · ·δ q kp k.By convention, the indices j, i 1 , i 2 , . . . , i λ run in the set {1, . . ., m}, theindices k, k 1 , k 2 , . . . , k λ and l, l 1 , l 2 , . . . , l λ running in {1, . . ., n}. Hence wewill write ∑ m ∑ mi 1 =1 i 2 =1 · · ·∑mi λ =1 as ∑ i 1 ,...,i λand ∑ n ∑ nl 1 =1 l 2 =1 · · ·∑nl λ =1as ∑ l 1 ,...,l λ. The letters i 1 , i 2 , . . .,i λ and l 1 , l 2 , . . .,l λ will always be used forthe summations in the development of R j k 1 ,k 2 ,...,k λ. We will always use theindices j and k 1 , k 2 , . . ., k λ to write the coefficient R j k 1 ,k 2 ,...,k λ.We have:(7.28)⎧R ⎪⎨j k 1= Rx j k1+ ∑ i 1⎪⎩∑ []δ l 1k1R j − u i 1 δj i 1Q l 1xk1U i 1l 1+l 1+ ∑ ∑i 1 ,i 2l 1 ,l 2[−δji 2δ l 1k1Q l 2u i 1]Ui 1l 1U i 2l 2.For R j k 1 ,k 2we have:(7.28) ⎧R j k 1 ,k 2= Rx j k1 x k2+ ∑ i 1+ ∑ ∑ [(δ l 1,l 2k 1 ,k 2R j u i 1u − i 2 δj i 2δ l 1k1Q l 2+ x k2 u i 1 δl 1k2Q l 2x k1 u i 1i 1 ,i 2 l 1 ,l 2⎪⎨+ ∑⎪⎩i 1 ,i 2 ,i 3∑ [−δ j i 3δ l 1,l 2k 1 ,k 2Q l 3u i 1u i 2l 1 ,l 2 ,l 3∑l 1[δ l 1k2R j x k1 u i 1 + δl 1k1R j x k2 u i 1 − δj i 1Q l 1xk1 x k2]U i 1l 1+]U i 1l 1U i 2l 2U i 3l 3++ ∑ ∑ []δ l 1,l 2k 1 ,k 2R j − u i 1 δj i 1δ l 1k2Q l 2xk1− δ j i 1δ l 1k1Q l 2xk2U i 1l 1 ,l 2+i 1 l 1 ,l 2+ ∑ i 1 ,i 2∑l 1 ,l 2 ,l 3[−δ j i 2δ l 1,l 2k 1 ,k 2Q l 3u i 1 − δj i 2δ l 3,l 1k 1 ,k 2Q l 2u i 1 − δj i 1δ l 2,l 3k 1 ,k 2Q l 1u i 2)]U i 1l 1U i 2l 2+]U i 1l 1U i 2l 2 ,l 3.Since we also treat systems of order κ ≥ 3, it is necessary to computeR j k 1 ,k 2 ,k 3. We write this as follows:(7.28) R j k 1 ,k 2 ,k 3= I + II + III,

where the first term I involves only polynomials in U i 1(7.28) ⎧I = Rx j k1 x k2 x k3+ ∑ ∑ [δ l 1k1R j + x k2 x k3 u i 1 δl 1k2R j + x k1 x k3 u i 1 δl 1k3R j − x k1 x k2 u i 1i 1 l 1]−δ j i 1Q l 1xk1 x k2 x k3U i 1l 1+ ∑ ∑ [δ l 1,l 2k 1 ,k 2R j x k3 u i 1u + i 2 δl 1,l 2k 3 ,k 1R j x k2 u i 1u + i 2i 1 ,i 2 l 1 ,l 2⎪⎨⎪⎩+δ l 1,l 2k 2 ,k 3R j x k1 u i 1u − i 2 δj i 2δ l 1k1Q l 2]−δ j i 2δ l 1k3Q l 2x k1 x k2U i u i 11 l 1U i 2l 2+ ∑−δ j i 3δ l 1,l 2k 1 ,k 2Q l 3−δ j i 3δ l 1,l 2k 1 ,k 3Q l 3+ ∑i 1 ,i 2 ,i 3 ,i 4x k3 u i 1u i 2 − δj i 3δ l 1,l 2x k2 u i 1u i 2x k2 x k3 u i 1 − δj i 2δ l 1i 1 ,i 2 ,i 3k 2 ,k 3Q l 3x k1 u i 1u − i 2]U i 1l 1U i 2l 2U i 3l 3+∑ [−δ j i 4δ l 1,l 2 ,l 3k 1 ,k 2 ,k 3Q l 4u i 1u i 2u i 3l 1 ,l 2 ,l 3 ,l 4l 1:k2Q l 2− x k1 x k3 u i 1∑ [δ l 1,l 2 ,l 3k 1 ,k 2 ,k 3R j u i 1u i 2u − i 3l 1 ,l 2 ,l 3]U i 1l 1U i 2l 2U i 3l 3U i 4l 4,163the second term II involves at least once the monomial U i 1l 1 ,l 2:(7.28)⎧II = ∑ i 1⎪⎨⎪⎩∑ [δ l 1,l 2k 1 ,k 2R j + x k3 u i 1 δl 1,l 2k 3 ,k 1R j + x k2 u i 1 δl 1,l 2k 2 ,k 3R j − x k1 u i 1l 1 ,l 2)]−δ j i 1(δ l 1k1Q l 2xk2 x k3+ δ l 1k2Q l 2xk1 x k3+ δ l 1k3Q l 2xk1 x k2U i 1l 1 ,l 2++ ∑ ∑ [δ l 1,l 2 ,l 3k 1 ,k 2 ,k 3R j u i 1u + i 2 δl 3,l 1 ,l 2k 1 ,k 2 ,k 3R j u i 1u + i 2 δl 2,l 3 ,l 1k 1 ,k 2 ,k 3R j u i 1u − i 2i 1 ,i 2 l 1 ,l 2 ,l 3−δ j i 1(δ l 2,l 3k 1 ,k 2Q l 1x k3 u i 2 + δl 2,l 3k 3 ,k 1Q l 1x k2 u i 2 + δl 2,l 3k 2 ,k 3Q l 1x k1 u i 2−δ j i 2(δ l 1,l 2k 1 ,k 2Q l 3x k3 u i 1 + δl 1,l 2k 3 ,k 1Q l 3x k2 u i 1 + δl 1,l 2k 2 ,k 3Q l 3+δ l 3,l 1k 1 ,k 2Q l 2+ ∑i 1 ,i 2 ,i 3+ x k3 u i 1 δl 3,l 1k 3 ,k 1Q l 2+ x k2 u i 1 δl 3,l 1k 2 ,k 3Q l 2x k1 u i 1x k1 u i 1 +)−)]U i 1l 1U i 2l 2 ,l 3+∑ [−δ j i 3(δ l 1,l 2 ,l 3k 1 ,k 2 ,k 3Q l 4u i 1u + i 2 δl 1,l 4 ,l 2k 1 ,k 2 ,k 3Q l 3u i 1u + i 2l 1 ,l 2 ,l 3 ,l 4) (− δ j i 1δ l 3,l 1 ,l 2k 1 ,k 2 ,k 3Q l 4u i 1u i 2+δ l 2,l 3 ,l 4k 1 ,k 2 ,k 3Q l 1u i 1u i 2δ l 3,l 2 ,l 4k 1 ,k 2 ,k 3Q l 1u i 2u i 3 + δl 4,l 3 ,l 2k 1 ,k 2 ,k 3Q l 1)]U i 1l 1U i 2l 2U i 3l 3 ,l 4+ ∑ i 1 ,i 2+δ l 3,l 1 ,l 2k 1 ,k 2 ,k 3Q l 4u i 1 + δl 2,l 3 ,l 1k 1 ,k 2 ,k 3Q l 4u i 1)]U i 1l 1 ,l 2U i 2l 3 ,l 4u i 2u i 3 +∑ [−δ j i 2(δ l 1,l 2 ,l 3k 1 ,k 2 ,k 3Q l 4+ u i 1l 1 ,l 2 ,l 3 ,l 4

164and the third term III involves at least once the monomial U i 1l 1 ,l 2 ,l 3(note thatthere is no term involving simultaneously U i 1l 1 ,l 2and U i 1l 1 ,l 2 ,l 3):(7.28) ⎧III = ∑ ∑ [(δ l 1,l 2 ,l 3k 1 ,k 2 ,k 3R j − u i 1 δj i 1δ l 1,l 2k 2 ,k 3Q l 3xk1+ δ l 1,l 2k 3 ,k 1Q l 3xk2+i 1 l 1 ,l 2 ,l 3⎪⎨)]+δ l 1,l 2k 1 ,k 2Q l 3xk3U i 1l 1 ,l 2 ,l 3+ ∑ ∑ [−δ j i 1δ l 2,l 3 ,l 4k 1 ,k 2 ,k 3Q l 1− u i 2i 1 ,i 2 l 1 ,l 2 ,l 3 ,l 4⎪⎩−δ j i 2(δ l 1,l 2 ,l 3k 1 ,k 2 ,k 3Q l 4u i 1 + δl 4,l 1 ,l 2k 1 ,k 2 ,k 3Q l 3u i 1 + δl 3,l 4 ,l 1k 1 ,k 2 ,k 3Q l 2u i 1)]U i 1l 1U i 2l 2 ,l 3 ,l 4.Before giving the partial expression of R κ we introduce some notations. Forp ∈ N with p ≥ 1, let S p be the group of permutations of {1, 2, . . ., p}. Forq ∈ N with 1 ≤ q ≤ p − 1, let S q p be the set of permutations σ ∈ S p suchthat σ(1) < σ(2) < · · · < σ(q) and σ(q + 1) < σ(q + 2) < · · · < σ(p). Itscardinal is C q p . Let C p be the group of cyclic permutations of {1, 2, . . ., p}.Reasoning recursively from the formula of R j k 1 ,k 2 ,k 3given by (7.28), we maygeneralize Lemma 8.1:Lemma 8.1. For every κ ≥ 4 and for every j = 1, . . ., m, k 1 , . . .,k κ =1, . . ., n, we have:(7.28) R j k 1 ,k 2 ,...,k κ= I 1 + · · · + I 9 + Remainderwhere I 1 = Rx j k1 x k2 ...x kκ,⎡I 2 = ∑ ∑⎣ ∑i 1 l 1 σ∈S 1 κ⎡∑⎣ ∑l 1 ,l 2I 3 = ∑ i 1I 4 = ∑ i 1σ∈S 2 κ−δ j i 1⎛⎝ ∑δ l 1kσ(1)R j x kσ(2)···x kσ(κ) u i 1 − δj i 1Q l 1xk1 ...x kκ⎤⎦U i 1l 1,δ l 1,l 2k σ(1) ,k σ(2)R j x kσ(3)···x kσ(κ) u i 1 −σ∈S 1 κ⎡∑⎣ ∑l 1 ,...,l κ−2σ∈S κ−2κ⎛−δ j ⎝ ∑i 1σ∈S κ−3κδ l 1kσ(1)Q l 2⎞xkσ(2)···x⎠kσ(κ)⎤⎦U i 1l 1 ,l 2,δ l 1,......,l κ−2k σ(1) ,...,k σ(κ−2)R j x kσ(κ−1) x kσ(κ) u i 1 −δ l 1,......,l κ−3k σ(1) ,...,k σ(κ−3)Q l κ−2⎞⎠x kσ(κ−2) x kσ(κ−1) x kσ(κ)⎤⎦U i 1l 1 ,...,l κ−2,

I 5 = ∑ i 1I 6 = ∑ i 1 ,i 2I 7 = ∑ i 1 ,i 2⎡∑⎣ ∑l 1 ,...,l κ−1σ∈S κ−1κδ l 1,......,l κ−1k σ(1) ,...,k σ(κ−1)R j x kσ(κ) u i 1 −⎛−δ j ⎝ ∑i 1σ∈S κ−2κ⎡∑⎣ ∑ δ l τ(1),...,l τ(κ)k 1 ,......,k κl 1 ,...,l κ τ∈C κ−δ j i 2⎛⎝ ∑σ∈S κ−1κ⎞δ l 1,......,l κ−2 l κ−1 ⎠k σ(1) ,...,k σ(κ−2) x kσ(κ−1) x kσ(κ)⎛R j u i 1u − i 2 δj ⎝ ∑i 1σ∈S κ−1κ⎤⎦U i 1165l 1 ,...,l κ−1,δ l 1,......,l κk σ(1) ,...,k σ(κ−1)Q l 1x kσ(κ) u i 2(δ l 1,......,l κ−1k σ(1) ,...,k σ(κ−1)Q lκx kσ(κ) u i 1 + · · · + δl 3,......,l 1k σ(1) ,...,k σ(κ−1)Q l 2x kσ(κ) u i 2× U i 1l 1U i 2l 2 ,...,l κ,∑ [ (−δ j i 1δ l 2,...,l κ+1k 1 ,...,k κQ l 1+ · · · + u i 2 δl κ+1,...,l 2k 1 ,...,k κQ l 1u i 2l 3 ,...,l κ+1⎛⎞⎤−δ j ⎝ ∑i 2δ l τ(1),...,l τ(κ)⎠⎦U i 1τ∈S 2 κk 1 ,......,k κQ l κ+1u i 1)−l 1 ,l 2U i 2l 3 ,...,l κ+1,⎡⎛⎞⎤I 8 = ∑ ∑⎣δ l 1,...,l κk 1 ,...,k κR j − u i 1 δj ⎝ ∑i 1δ l 1,......,l κ−1k σ(1) ,...,k σ(κ−1)Q lκ ⎠x kσ(κ)⎦U i 1i 1 l 1 ,...,l κ σ∈Sκκ−1I 9 = ∑ ∑ [(−δ j i 1δ l 2,...,l κ+1k 1 ,...,k κQ l 1− u i 2 δj i 2δ l 1,...,l κk 1 ,...,k κQ l κ+1+ · · · + δ l 3,...,l 1u i 1i 1 ,i 2 l 1 ,...,l κ+1l 1 ,...,l κ,k 1 ,...,k κQ l 2u i 1× U i 1l 1U i 2l 2 ,...,l κ+1and where the term Remainder denotes the remaining terms in the expansionof R j k 1 ,k 2 ,...,k κ.In I 6 the summation on the upper indices (l 1 , . . .,l κ ) gets on all the circularpermutations of {1, 2, . . ., κ} except the identity. In I 7 the summationgets on all the circular permutations of {2, 3, . . ., κ + 1}. In I 9 the summationgets on all the circular permutations of {1, 2, . . ., κ + 1} except theone transforming (l 1 , l 2 , . . .,l κ+1 ) into (l 2 , l 3 , . . .,l 1 ). For κ = 3, comparingwith (7.28), we see that the formula remains valid, with the same conventionsas in the case n = 1.3.4. Lie criterion and defining equations of Sym(E ). We recall the Liecriterion, presented in Subsection 2.6 (see Theorem 2.71 of [Ol1986]):A vector field X is an infinitesimal symmetry of the completely integrablesystem (E ) if and only if its prolongation X (κ) of order κ is tangent to theskeleton ∆ E in the jet space J κ n,m .⎞⎠ −) ⎞ ⎤⎠⎦ ×)]×

166The set of infinitesimal symmetries of (E ) forms a Lie algebra, since wehave the relation [X, X ′ ] (κ) = [X (κ) , X ′ (κ) ] (cf. [Ol1986]). We will denoteby Sym(E ) this Lie algebra. The aim of the forecoming Section is to obtainprecise bounds on the dimension of the Lie algebra Sym(E ) of infinitesimalsymmetries of (E ). For simplicity we start with the case n = m = 1.4. OPTIMAL UPPER BOUND ON DIM K Sym(E ) WHEN n = m = 1.4.1. Defining equations for Sym(E ). Applying the Lie criterion, the tangencycondition of X (κ) to ∆ E is equivalent to the identity:(7.28)R κ −[Q ∂F∂x + R ∂F∂u∂F ∂F+ R1 + R2∂U1 ∂U + · · · + 2 Rκ−1]∂F≡ 0,∂U κ−1on the subvariety ∆ E , that is to a formal identity in K{x, u, U 1 , . . ., U κ−1 },in which we replace the variable U κ by F(x, u, U 1 , . . .,U κ−1 ) in the twomonomials U κ and U 1 U κ of R κ , cf. Lemma 8.1. Expanding F and its partialderivatives in power series of the variables (U 1 , . . .,U κ−1 ) with analyticcoefficients in (x, u), we may rewrite (7.28) as follows:⎧ ∑ [⎪⎨ Φµ1 ,...,µ κ−1(x, u, (Q x k u l) k+l≤κ, (R x k u l) k+l≤κ) ] ×(7.28) µ 1 ,...,µ κ−1 ≥0⎪⎩×(U 1 ) µ 1. . .(U κ−1 ) µ κ−1≡ 0,where the expressions(7.28) Φ µ1 ,...,µ κ−1(x, u, (Q x k u l) k+l≤κ, (R x k u l) k+l≤κ)are linear with respect to the partial derivatives ((Q x k u l) k+l≤κ, (R x k u l) k+l≤κ),with analytic coefficients in (x, u). By construction these coefficients essentiallydepend on the expansion of F . The tangency condition (7.28)is equivalent to the following infinite linear system of partial differentialequations, called defining equations of Sym(E ):(7.28) Φ µ1 ,...,µ κ−1(x, u, (Q x k u l(x, u)) k+l≤κ, (R x k u l(x, u)) k+l≤κ) = 0,satisfied by (Q(x, u), R(x, u)). The Lie method consists in studying thesolutions of this linear system of partial differential equations.4.2. Homogeneous system. As mentioned in the introduction, we focusour attention on the case κ ≥ 3. Denote by (E 0 ) the homogeneous equationu x κ = 0 of order κ. The general solution u = ∑ κ−1l=0 λ l x l consists ofpolynomials of degree ≤ κ − 1 and the defining equation (7.28) reduces toR κ = 0. Using the expression (7.28), expanding (7.28), (7.28) and consideringonly the coefficients of the five monomials ct., U κ−2 , U κ−1 , U 1 U κ−1 and

U 2 U κ−1 , we obtain the five following partial differential equations, whichare sufficient to determine Sym(E 0 ):⎧R x κ = 0,(κ − 2)R x ⎪⎨2 u − Q x 3 = 0,3(7.28)(κ − 1)R xu − Q x 2 = 0,2R u 2 − κ Q xu = 0,⎪⎩Q u = 0.The general solution of this system is evidently:167(7.28){Q = A + B x + C x 2 ,R = (κ − 1) C xu + D u + E 0 + E 1 x + · · · + E κ−1 x κ−1 ,where the (κ + 4) constants A, B, C, D, E 0 , E 1 , . . ., E κ−1 are arbitrary.Computing explicitely the flows of the (κ + 4) generators ∂/∂x, x∂/∂x,x 2 ∂/∂x + (κ − 1) xu ∂/∂u, u ∂/∂u, ∂/∂u, x∂/∂u, . . . , x κ−1 ∂/∂u, wecheck easily that they stabilize the graphs of polynomials of degree ≤ κ −1.Moreover they span a Lie algebra of dimension (κ+4) and the general formof a Lie symmetry is:( )α0 + α 1 x(7.28) (x, u) ↦−→1 + εx , βu + γ 0 + γ 1 x + · · · + γ κ−1 x κ−1.(1 + εx) κ−14.3. Nonhomogeneous system. Consider for κ ≥ 3 the equation (7.28) afterreplacing the variable U κ by F . Let Φ(U λ ) denote an arbitrary termof the form φ(x, u) U λ , where φ(x, u) is an analytic function. We considerthe five following terms Φ(ct.), Φ(U κ−2 ), Φ(U κ−1 ), Φ(U 1 U κ−1 )and Φ(U 2 U κ−1 ). Since some multiplications of monomials appear inthe expression (7.28), we must be aware of the fact that Φ(U 1 U κ−1 ) ≡Φ(U 1 ) Φ(U κ−1 ) and Φ(U 2 U κ−1 ) ≡ Φ(U 2 ) Φ(U κ−1 ). Consequently in theexpansion of (7.28) we must take into account the seven types of monomialsΦ(ct.), Φ(U 1 ), Φ(U 2 ), Φ(U κ−2 ), Φ(U κ−1 ), Φ(U 1 U κ−1 ) and Φ(U 2 U κ−1 ).The (κ + 1) derivatives ∂F/∂x, ∂F/∂u, ∂F/∂U 1 , . . .,∂F/∂U κ−1 appearingin the brackets of (7.28), and the term F appearing in the expressionof R κ after replacing U κ by F (cf. the last two monomials U κ and U 1 U κin (7.28)) may all contain the seven monomials ct., U 1 , U 2 , U κ−2 , U κ−1 ,U 1 U κ−1 and U 2 U κ−1 . For F and its (κ + 1) first derivatives we use thegeneric simplified notation(7.28)Φ(ct.)+Φ(U 1 )+Φ(U 2 )+Φ(U κ−2 )+Φ(U κ−1 )+Φ(U 1 U κ−1 )+Φ(U 2 U κ−1 ),

168to name the seven monomials appearing a priori. Hence, expanding (7.28),picking up the only terms which may contain the five monomials we areinterested in, and using the formula of Lemma 8.1 for R λ (1 ≤ λ ≤ κ), weobtain the following expression:(7.28)⎧R x κ + [ Cκ 2 R x 2 u − Cκ 3 Q ]x 3 U κ−2 + [ Cκ 1 R xu − Cκ 2 Q ]x 2 U κ−1 +⎪⎨⎪⎩+ [ C 1 κ R u 2 − κ 2 Q xu]U 1 U κ−1 + [ −C 2 κ+1 Q u]U 2 U κ−1 ++ { R u − C 1 κ Q x + [ −C 1 κ+1 Q u]U1 } ×× { Φ(ct.) + Φ(U 1 ) + Φ(U 2 ) + Φ(U κ−2 ) + Φ(U κ−1 ) + Φ(U 1 U κ−1 ) + Φ(U 2 U κ−1 ) } −− { Q + R + R x + [R u − Q x ] U 1 + R x 2 + [2R xu − Q x 2] U 1 ++ [R u − 2Q x ] U 2 + · · · + R x κ−3 + [ C 1 κ−3 R x κ−4 u − Q x κ−3]U 1 ++ [ C 2 κ−3 R x κ−5 u − C 1 κ−3 Q x κ−4]U 2 + R x κ−2++ [ C 1 κ−2 R x κ−3 u − Q x κ−2]U 1 + [ C 2 κ−2 R x κ−4 u − C 1 κ−2 Q x κ−3 ]U 2 ++ [ R u − C 1 κ−2 Q x]U κ−2 + [ −C 1 κ−1 Q u]U 1 U κ−2 ++R x κ−1 + [ C 1 κ−1 R x κ−2 u − Q x κ−1]U 1 ++ [ C 2 κ−1 R x κ−3 u − C 1 κ−1 Q x κ−2 ]U 2 + [ C 1 κ−1 R xu − C 2 κ−1 Q x 2 ]U κ−2 ++ [ C 1 κ−1 R u 2 − (κ − 1)2 Q xu]U 1 U κ−2 + [ R u − C 1 κ−1 Q x]U κ−1 ++ [ −C 1 κ Q u]U 1 U κ−1} ×× { Φ(ct.) + Φ(U 1 ) + Φ(U 2 ) + Φ(U κ−2 ) + Φ(U κ−1 ) + Φ(U 1 U κ−1 ) + Φ(U 2 U κ−1 ) }+ Remainder ≡ 0.Here the term Remainder consists of the monomials, in the jet variables,different from the five ones we are concerned with. The first four lines beforethe sign “−” develop R κ and the third line consists of the factor F replacedby (7.28). In the last line (note that this is multiplied by the nine precedinglines) we replaced the (κ+1) first partial derivatives of F appearing in (7.28)by the term (7.28) which we factorized.By expanding the product appearing in this expression (7.28), and equalingto zero the coefficients of the five monomials ct., U κ−2 , U κ−1 , U 1 U κ−1

and U 2 U κ−1 , we obtain the five following partial differential equations(7.28)⎧R x κ = Π(x, u, Q, Q x , R, R x , . . .,R x κ−1, R u ),⎪⎨⎪⎩C 2 κ R x 2 u − C 3 κ Q x 3 = Π(x, u, Q, Q x, Q x 2, R, R x , . . .,R x κ−1, R u , R xu ),C 1 κ R xu − C 2 κ Q x 2 = Π(x, u, Q, Q x, R, R x , . . .,R x κ−1, R u ),C 1 κ R u 2 − κ 2 Q xu = Π(x, u, Q, Q x , . . .,Q x κ−1, Q u , R, R x , . . .R x κ−1,169R u , R xu , . . .,R x κ−2 u),−C 2 κ+2 Q u = Π(x, u, Q, Q x , . . .,Q x κ−2, R, R x , . . .R x κ−1,R u , R xu , . . .,R x κ−3 u).Here by convention Π denotes any linear quantity in Q, R and some of theirderivatives, of the form⎧Π(x, u, Q ⎪⎨ x a 1u b 1 , . . ., Q x ap u bp, R x c 1u 1, . . ., R d x cq u dq) =(7.28)p∑q∑⎪⎩ = φ i (x, u) Q x a iu b i (x, u) + ψ j (x, u) R xc j u j(x, u), di=1where φ i and ψ j are analytic in (x, u). For instance, the differentiationof Π(x, u, Q, R, R u ) with respect to x gives the expressionΠ(x, u, Q, Q x , R, R x , R xu ). Let us introduce the followingcollection of (κ + 4) partial derivatives of (Q, R) defined byJ := (Q, Q x , Q x 2, R, R x , . . ., R x κ−1, R u ). The aim is now to makelinear substitutions on the system (7.28) to obtain the system (7.28) wherethe five second members depend only on the collection J. The desiredestimate dim K Sym(E ) ≤ κ + 4 will follow from (7.28).Let us differentiate the third equation of (7.28) with respect to x. Dividingby Cκ 1 we obtain:(7.28)(κ − 1)R x 2 u − Q x 3 = Π(x, u, Q, Q x , Q x 2, R, R x , . . .,R x κ, R u , R xu ).2Solving R x 2 u and Q x 3 by the second equality in (7.28) and by (4.3) we find{Qx 3 = Π(x, u, Q, Q x , Q x 2, R, R x , . . ., R x κ, R u , R xu ),(7.28)R x 2 u = Π(x, u, Q, Q x , Q x 2, R, R x , . . ., R x κ, R u , R xu ).Replacing R x κ by its value given by the first equality in (7.28) we obtain forQ x 3:(7.28) Q x 3 = Π(x, u, Q, Q x , Q x 2, R, R x , . . .,R x κ−1, R u , R xu ).If we write the third equality in (7.28) as(7.28) R xu = Π(x, u, Q, Q x , Q x 2, R, R x , . . .,R x κ−1, R u ),j=1

170we may replace R xu in (7.28). This gives the desired dependence of Q x 3 onthe collection J:(7.28) Q x 3 = Π(x, u, Q, Q x , Q x 2, R, R x , . . .,R x κ−1, R u ).We may now differentiate the equalities (7.28) and (7.28) with respect to xup to the order l. At each differentiation we replace Q x 3, R xu and R x κ bytheir values in (7.28), in (7.28) and in the first equality in (7.28) respectively.We obtain for l ∈ N:{Qx l+3 = Π(x, u, Q, Q x , Q x 2, R, R x , . . .,R x κ−1, R u ),(7.28)R x l+1 u = Π(x, u, Q, Q x , Q x 2, R, R x , . . .,R x κ−1, R u ).Replacing these values in the fifth equality of (7.28), we obtain(7.28) Q u = Π(x, u, Q, Q x , Q x 2, R, R x , . . ., R x κ−1, R u ).By replacing the fourth equality of (7.28) we obtain finally(7.28) R u 2 = Π(x, u, Q, Q x , Q x 2, R, R x , . . .,R x κ−1, R u ).To summarize, using the first equality of (7.28), using (7.28), (7.28), (7.28)and (7.28), we obtained the desired system:⎧R x κ = Π(x, u, Q, Q x , Q x 2, R, R x , . . .,R x κ−1, R u ),(7.28)⎪⎨Q u = Π(x, u, Q, Q x , Q x 2, R, R x , . . .,R x κ−1, R u ),R u 2 = Π(x, u, Q, Q x , Q x 2, R, R x , . . .,R x κ−1, R u ),R xu = Π(x, u, Q, Q x , Q x 2, R, R x , . . .,R x κ−1, R u ),⎪⎩Q x 3 = Π(x, u, Q, Q x , Q x 2, R, R x , . . .,R x κ−1, R u ).We recall that the terms Π are linear expressions of the form (7.28). Let usdifferentiate every equation of system (7.28) with respect to x at an arbitraryorder and let us replace in the right hand side the terms R x κ, R xu and Q x 3 thatmay appear at each step by their value in (7.28), and then differentiate withrespect to u at an arbitrary order. We deduce that all the partial derivativesof the five functions R x κ, Q u , R u 2, R xu and Q x 3 are also linear functionsof the (κ + 4) partial derivatives (Q, Q x , Q x 2, R, R x , . . .,R x κ−1, R u ). Thusthe analytic functions Q and R are determined uniquely by the value at theorigin of the (κ + 4) partial derivatives (Q, Q x , Q x 2, R, R x , . . .,R x κ−1, R u ).This ends the proof of the inequality dim K Sym(E ) ≤ κ + 4.5. OPTIMAL UPPER BOUND ON DIM K Sym(E ) IN THE GENERALDIMENSIONAL CASE5.1. Defining equations for Sym(E ). In the general dimensional case, thetangency condition of the prolongation X κ of X to the skeleton gives the

following equations for j = 1, . . ., m and k 1 , . . .,k κ = 1, . . .,n:(7.28)⎧ [ n∑R j k⎪⎨1 ,...,k κ− Q ∂F j m∑l k 1 ,...,k κ+ R ∂F j i k 1 ,...,k κ+∂x l ∂u i l=1i=1⎤⎪⎩+ ∑ ∑ ∂F jR i 1 k 1 ,...,k κl1i 1∂U i + · · · + ∑ ∑ ∂F jR i 1 k 1 ,...,k κ1l1⎦,...,l κ−1l 1 l 1 i 1∂U i ≡ 0,1l 1 ,...,l κ−1 l 1 ,...,l κ−1on ∆ E , by replacing the variables U i 1l 1 ,...,l κby F i 1l 1 ,...,l κwherever they appear.Let us expand F j k 1 ,...,k κand their partial derivatives and use the fact thatR j k 1 ,...,k λare polynomials expressions of the jets variables (U i 1l 1, . . .,U i 1l 1 ,...,l λ),with coefficients being linear expressions of the partial derivatives of order≤ λ+1 of Q l and R j . We obtain for j = 1, . . ., m and k 1 , . . .,k κ = 1, . . .,nsome identities of the form(7.28) ⎧ ∑⎪⎨⎪⎩i 1 , ...,l 1 ,...)Φ j;i 1,......k 1 ,...,k κ; l 1 ,......(x,u,(Q l x α u) β 1≤l≤n, |α|+|β|≤κ+1 ,(R j )x α u β 1≤j≤m, |α|+|β|≤κ+1 ×171×U i 1l 1... U iµ 1l µ1× U iµ 1 +1l µ1 +1,l µ1 +2 · · · Ui µ 1 +µ 2 −1l µ1 +2µ 2 −1 Ui µ 1 +µ 2l µ1 +2µ 2× · · · · · · ≡ 0,satisfied if and only if the functions Q l and R j are solutions of the followingsystem of partial differential equations(7.28)Φ j,i (1,......k 1 ,...,k κ; l 1 ,...... x, u, (Qlx α u) β 1≤l≤n, |α|+|β|≤κ+1 , (R j ))x α u β 1≤j≤m, |α|+|β|≤κ+1 = 0.5.2. Homogeneous system. We start by giving the general form of thesymmetries of the homogeneous system in the case κ = 2. Then we provethe equality dim K (Sym(E 0 )) = n 2 +2n+m 2 +m Cn+κ−1 in the case κ ≥ 3.In the case κ = 2 we obtain:⎧n∑m∑Q l (x, u) = A l + Bk l 1x k1 + Ci l 1u i 1+(7.28)⎪⎨⎪⎩R j (x, u) = F j +k 1 =1n∑k 1 =1+n∑k 1 =1G j k 1x k1 ++n∑k 1 =1i 1 =1D k1 x l x k1 +m∑E i1 x l u i 1,i 1 =1m∑H j i 1u i 1+i 1 =1D k1 x k1 u j +m∑E i1 u i 1u j .i 1 =1

172Here the (n+m)(n+m+2) constants A l , B l k 1, C l i 1, D k1 , E i1 , F j , G j k 1, H j i 1∈K are arbitrary. Moreover one can check that the vector space spanned bythe (n + m)(n + m + 2) vector fields(7.28)⎧(⎪⎨ x k1⎪⎩∂∂x k1, x k1u i 1x 1(x 1∂∂u i 1 , x k 1∂∂x k1,∂, u i 1∂x k2∂+ · · · + x n∂x 1∂∂x 1+ · · · + x n∂∂u i 1 , ui 1∂∂x n+ u 1∂+ u 1∂x n∂∂u i 2∂∂u 1 + · · · + um∂∂u 1 + · · · + um∂ ),∂u m∂ ),∂u mis stable under the Lie bracket action and that the flow of each of thesegenerators is a Lie symmetry of the system (E 0 ). This proves that Sym(E 0 )is indeed a Lie algebra with dimension (n + m)(n + m + 2). Finally thecorresponding transformations close to the identity mapping are projective,represented by the formula:(7.28)⎧(x, u) ↦−→ ⎪⎨⎪⎩( (αl,0+ ∑ nk=1 α l,k x k + ∑ mi=1 α l,n+i u ii=1 γ ,n+i u)1≤l≤ni )1 + ∑ nk=1 γ k x k + ∑ m(βj,0 + ∑ nk=1 β j,k x k + ∑ mi=1 β j,n+i u i1 + ∑ nk=1 γ k x k + ∑ mi=1 γ n+i u i )1≤j≤m.It is clear that these transformations preserve all the solutions of (E 0 ) :u j x k1 x k2= 0, the graphs of affine maps from K n to K m .In the case κ ≥ 3 we consider the homogeneous system (E 0 ) in which thesecond members F j k 1 ,...,k κvanish identically. Its solutions are the graphs ofpolynomial maps of degree ≤ (κ − 1) from K n to K m . The defining equationsof its Lie algebra of infinitesimal symmetries are R j k 1 ,...,k κ= 0, afterhaving replaced the variables U i 1l 1 ,...,l κby 0 = F i 1l 1 ,...,l κin I 8 and I 9 in (7.28).We will keep in this system the only equations coming from the vanishingof the coefficients of the five families of monomials ct., U i 1l 1 ,...,l κ−2, U i 1l 1 ,...,l κ−1,U i 1l 1U i 2l 2 ,...,l κand U i 1l 1 ,l 2U i 2Subsection 4.2). The coefficients of these five monomials families alreadyl 3 ,...,l κ+1(this is inspired from the computations inappear in the expression (7.28). Moreover we fix l 1 = l 2 = · · · = l κ+1 = land i 1 = i 2 , except for the coefficient of the monomial U i 1lU i 2l,...,l, where wefix first i 1 = i 2 and then i 1 ≠ i 2 . This provides the six partial differential

linear equations:(7.28) ⎧0 = Rx j k1 x k2···x kκ,0 = ∑δ l,.........,lk σ(1) ,...,k σ(κ−2)R j −x kσ(κ−1) x kσ(κ) u i 1⎪⎨⎪⎩σ∈S κ−2κ0 = ∑σ∈S κ−1κ− δ j i 1⎛⎝ ∑σ∈S κ−3κδ l,.........,lk σ(1) ,...,k σ(κ−1)R j x kσ(κ) u i 1 −− δ j i 1⎛⎝ ∑σ∈S κ−2κ0 = κ δ l,......,lk 1 ,...,k κR j u i 1u i 1 − κ δj i 1⎛⎝ ∑0 = 2κ δ l,......,lk 1 ,...,k κR j u i 1u i 2 − κ δj i 1⎛σ∈S κ−1κ⎝ ∑− κ δ j i 2⎛⎝ ∑0 = − C 2 κ+1 δ j i 1δ l,......,lk 1 ,...,k κQ l u i 1.173δ l,.........,lk σ(1) ,...,k σ(κ−3)Q l x kσ(κ−2) x kσ(κ−1) x kσ(k)⎞⎠,δ l,.........,lk σ(1) ,...,k σ(κ−2)Q l x kσ(κ−1) x kσ(κ)⎞⎠ ,σ∈S κ−1κσ∈S κ−1κδ l,.........,lk σ(1) ,...,k σ(κ−1)Q l x kσ(κ) u i 1⎞δ l,.........,lk σ(1) ,...,k σ(κ−1)Q l x kσ(κ) u i 2δ l,.........,lk σ(1) ,...,k σ(κ−1)Q l x kσ(κ) u i 1⎞⎠ ,⎞⎠ −⎠ , i 1 ≠ i 2 ,To solve the system (7.28) we fix the indices k 1 = · · · = k κ = l and j = i 1 inthe sixth equation, implying Q l = 0. Hence the terms following u i 1 δj i 1and δ j i 2in the fourth and in the fifth equations vanish identically. Let us choose theindices k 1 = · · · = k κ in the fourth and the fifth equations (this last equationis satisfied only for i 1 ≠ i 2 ). We obtain first three simple equations, withoutany restriction on the indices:⎧0 = Rx ⎪⎨j k1 x k2 ···x kκ,(7.28)0 = Q l u 1,⎪⎩i 0 = R j u i 1u . i 2Finally we specify the indices in the third equation of (7.28) as follows:l = k κ = · · · = k 3 = k 2 = k 1 ; then l = k κ = · · · = k 3 = k 2 ≠ k 1 ;finally l = k κ = · · · = k 3 , k 3 ≠ k 2 , k 3 ≠ k 1 . This gives the three following

174equations:(7.28)⎧⎪⎨⎪⎩0 = C 1 κ Rj x k1 u i 1 − C2 κ δj i 1Q k 1x k1 x k1,0 = R j x k1 u i 1 − C1 κ−1 δ j i 1Q k 2x k1 x k2, k 2 ≠ k 1 ,0 = − δ j i 1Q k 3x k1 x k2, k 3 ≠ k 1 , k 3 ≠ k 2 .We specify the indices in the second equation of (7.28) as follows: l =k κ = · · · = k 3 = k 2 = k 1 ; then l = k κ = · · · = k 3 = k 2 ≠ k 1 ; thenl = k κ = · · · = k 3 , k 3 ≠ k 2 , k 3 ≠ k 1 ; finally l = k κ = · · · = k 4 , l ≠ k 1 ,l ≠ k 2 , l ≠ k 3 . This gives the four following equalities:(7.28)⎧⎪⎨0 = C 2 κ R j x k1 x k1 u i 1 − C3 κ δ j i 1Q k 1x k1 x k1 x k1,0 = C 1 κ−1 R j x k1 x k2 u i 1 − C2 κ−1 δ j i 1Q k 2x k1 x k2 x k2, k 2 ≠ k 1 ,0 = R j − x k1 x k2 u i 1 C1 κ−2 δj i 1Q k 3x k1 x k2 x k3, k 3 ≠ k 1 , k 3 ≠ k 2 ,⎪⎩ 0 = − δ j i 1Q l x k1 x k2 x k3, l ≠ k 1 , l ≠ k 2 , l ≠ k 3 .Let us differentiate now the equations (7.28) with respect to the variables x las follows: we differentiate (7.28) 1 with respect to x k1 ; then we differentiate(7.28) 2 with respect to x k2 ; finally we differentiate (7.28) 3 with respect tox k3 . This gives the three following equations:(7.28)⎧⎪⎨⎪⎩0 = C 1 κ Rj x k1 x k1 u i 1 − C2 κ δj i 1Q k 1x k1 x k1 x k1,0 = R j x k1 x k2 u i 1 − C1 κ−1 δj i 1Q k 2x k1 x k2 x k2, k 2 ≠ k 1 ,0 = − δ j i 1Q k 3x k1 x k2 x k3, k 3 ≠ k 1 , k 3 ≠ k 2 .The seven equations given by the systems (7.28) and (7.28) may be consideredas three systems of two equations (of two variables) with a nonzerodeterminant, to which we add the last equation (7.28) 4 . We get immediately:(7.28)⎧⎪⎨0 = R j x k1 x k1 u i 1 = δj i 1Q k 1x k1 x k1 x k1,0 = R j x k1 x k2 u i 1 = δj i 1Q k 2x k1 x k2 x k2, k 2 ≠ k 1 ,0 = R j = x k1 x k2 u i 1 δj i 1Q k 3x k1 x k2 x k3, k 3 ≠ k 1 , k 3 ≠ k 2 ,⎪⎩ 0 = δ j i 1Q l x k1 x k2 x k3, l ≠ k 1 , l ≠ k 2 , l ≠ k 3 .It follows from these relations and from the relations Q l u i 1 = Rj u i 1u i 2 = 0obtained in (7.28) that all the third order partial derivatives of Q l vanishidentically, this being also satisfied by the third order partial derivatives of

175R j containing at least one partial derivative with respect to u i 1:⎧⎨ 0 = Q l x k1 x k2 x k3= Q l x k1 x k2 u(7.28)i 1= Q l x k1 u i 1u i 2= Q l u i 1u i 2u 3, i⎩ 0 = R j = x k1 x k2 u i 1 Rj x k1 u i 1u = i 2 Rj u i 1u i 2u . i 3It follows from the equations (7.28) and (7.28) that all the functions Q l arepolynomials of degree ≤ 2 with respect to the variables x k1 and all the functionsR j are a sum of a polynomial of degree ≤ (κ − 1) in the variables x k1and of monomials of the form u i 1and x k1 u i 1. Let us develop now the relations(7.28) separately for j = i 1 and j ≠ i 1 . We obtain the five equations:(7.28)⎧0 = Cκ ⎪⎨1 Rj , j ≠ i x k1 u i 1 1,0 = R i 1− x k1 u i 1 C1 κ−1 Q k 2x k1 x k2, k 2 ≠ k 1 ,⎪⎩0 = C 1 κ R i 1x k1 u i 1 − C2 κ Q k 1x k1 x k1,0 = R j x k1 u i 1 , j ≠ i 1,0 = − Q k 3x k1 x k2, k 3 ≠ k 1 , k 3 ≠ k 2 .According to the equations (7.28), (7.28), (7.28), we have the followingform of the general solution:(7.28) ⎧n∑n∑Q l (x, u) = A l + Bk l 1x k1 + C k1 x k1 x l ,⎪⎨R j (x, u) =⎪⎩k 1 =1k 1 =1n∑(κ − 1) C k1 x k1 u j +k 1 =1+ · · · +∑1≤k 1 ≤···≤k κ−1 ≤nm∑D j i 1u i 1+ E j,0 +i 1 =1E j,κ−1k 1 ,...,k κ−1x k1 · · ·x kκ−1 .n∑k 1 =1E j,1k 1x k1 +Here the n+n 2 +n+m 2 +m Cn+κ−1 constants A l , Bk l 1, C k1 , D j i 1, E j,0 , E j,1k 1,. . . , E j,κ−1k 1 ,...,k κ−1∈ K are arbitrary. Moreover one can check that the vectorspace spanned by the vector fields(7.28)⎧⎪⎨∂∂x k1, x k1∂,∂x k2∂+ · · · + x n∂x 1((∂x k1 x 1+ (κ − 1) u 1 ∂∂x n ∂u + · · · + ∂ ))1 um ,∂u m⎪⎩ u i ∂1∂u , ∂i 2 ∂u , x ∂i k 11 ∂u , . . ...., x ∂i k 1· · ·x kκ−11 ∂u , i 1is stable under the Lie bracket action and that the flow of each of thesegenerators is indeed a Lie symmetry of the system (E 0 ). Finally the Lie

176symmetries of (E 0 ) have the following form:(7.28) ( (αl,0+ ∑ nk=1(x, u) ↦−→α l,k x k1 + ∑ nk=1 ε ,k x k)1≤l≤n(∑ mi 1=1 βj i 1u i1 + γ 0,j + ∑ nk 1=1 γ1,j k 1x k1 + · · · + ∑ ⎞k 1≤···≤k κ−1γ κ−1,jk 1,...,k κ−1x k1 · · · x kκ−1[1 + ∑ nk=1 ε ⎠.k x k ])1≤j≤mκ−1We note again that these transformations preserve the solutions of (E 0 ) :u j x k1···x kκ= 0, namely the graphs of polynomial maps of degree ≤ (κ − 1)from K n to K m .5.3. Nonhomogeneous system. Let κ ≥ 3. Let us expand the definingequations (7.28) as done in (7.28). We will write only the coefficientsof the five monomial families ct., U i 1l 1 ,...,l κ−2, U i 1l 1 ,...,l κ−1, U i 1l 1U i 2l 2 ,...,l κandU i 1l 1 ,l 2U i 2l 3 ,...,l κ+1. Moreover, we fix always l 1 = l 2 = · · · = l κ = l κ+1 = land i 1 = i 2 , except for the fourth family of monomials where we distinguishthe two cases i 1 = i 2 and i 1 ≠ i 2 . Thus we obtain six linear equations ofpartial derivatives, the members on the left side (coming from the expressionof R j 1k1 ,...,k κgiven by Lemma 8.1) coincide with the members on theright hand side of (7.28). Furthermore, the members on the right hand sideare exactly the same as those obtained in (7.28), with more indices! We usethe letters l ′ , k 1 ′ , . . ., k′ κ = 1, . . .,n and j′ , i ′ 1 = 1, . . ., m for the indices ofthe arguments of the expressions Π, obtaining the six following equations,which generalize the equations (7.28):(7.28)[1] : R j x k1 x k2 ···x kκ= Π(x, u, Q l′ , Q l′x k ′1, R j′ , R j′x k ′1, . . .,R j′x k ′ 1···x k ′κ−1, R j′u i′ 1),([1] : Rx j k1 x k2···x kκ= Π x, u, Q l′ , Q l′x k, R j′ , R′ x j′k, . . .,R′ x j′k ′1 1 1···x k, R j′′κ−1∑[2] : δ l,.........,lk σ(1) ,...,k σ(κ−2)R j −x kσ(κ−1) x kσ(κ) u i 1σ∈S κ−2κ− δ j i 1⎛⎝ ∑(= Π x, u, Q l′ , Q l′x k, Q l′′1σ∈S κ−3κδ l,.........,lk σ(1) ,...,k σ(κ−3)Q l x kσ(κ−2) x kσ(κ−1) x kσ(k)⎞⎠ =x k ′ x k, R j′ , R′ x j′k, . . .,R′ x j′k ′1 2 1 1···x k, R j′′κ−1u i′ 1u i′ 1), R j′.x k ′ u i′ 11).

177[3] :∑σ∈S κ−1κδ l,.........,lk σ(1) ,...,k σ(κ−1)R j x kσ(κ) u i 1 −(= Π x, u, Q l′ , Q l′x k, R j′ , R j′′1− δ j i 1⎛⎝ ∑σ∈S κ−2κ[4] : κ δ l,......,lk 1 ,...,k κR j u i 1u i 1 − κ δj i 1⎛⎝ ∑(= Π x, u, Q l′ , Q l′x k, . . .,Q l′′1δ l,.........,lk σ(1) ,...,k σ(κ−2)Q l x kσ(κ−1) x kσ(κ)⎞⎠ =x k, . . .,R′ x j′k ′1 1···x k, R j′′κ−1σ∈S κ−1κu i′ 1).δ l,.........,lk σ(1) ,...,k σ(κ−1)Q l x kσ(κ) u i 1x k ′ ···x 1 k, Q l′, , R′κ−1 u i′ 1 Rj′ x j′k, . . .,R j′′1, R j′, R j′, . . ., R j′u i′ 1 x k ′ u i′ 1 x 1 k ′ ···x 1 k ′ u i′ 1κ−2[5] : 2κ δ l,......,lk 1 ,...,k κR j u i 1u i 2 − κ δj i 1⎛⎝ ∑− κ δ j i 2⎛⎝ ∑(= Π x, u, Q l′ , Q l′x k, . . .,Q l′′1σ∈S κ−1κσ∈S κ−1κ)δ l,.........,lk σ(1) ,...,k σ(κ−1)Q l x kσ(κ) u i 2δ l,.........,lk σ(1) ,...,k σ(κ−1)Q l x kσ(κ) u i 1.⎞x k ′ 1···x k, Q l′, , R′κ−1 u i′ 1 Rj′ x j′k, . . .,R j′′1, R j′, R j′, . . .,R j′u i′ 1 x k ′ u i′ 1 x k ′1 1···x k ′ u i′ 1κ−2).⎞⎠ =x k ′ 1···x k ′κ−1,⎞⎠ −⎠ , i 1 ≠ i 2 .x k ′1 ···x k ′κ−1,[6] : −Cκ+1 2 δj i 1δ l,......,lk 1 ,...,k κQ l u i 1=(= Π x, u, Q l′ , Q l′x k, . . .,Q l′′ x 1 k ′ 1···x k, R j′ , R′ x j′κ−2k, . . ., R j′′1)Then we get the following Lemma:, R j′, R j′, . . .,R j′u i′ 1 x k ′ u i′ 1 x k ′1 1···x k ′ u i′ 1κ−3.x k ′1 ···x k ′κ−1,

178Lemma 8.1. Let J denote the collection of n + n 2 + n + m Cn+κ−1 + m 2partial derivatives(7.28) J :=(Q l′ , Q l′x k ′1, Q k′ 1x k ′1x k ′1, R j′ , R j′x k ′1, . . .,R j′x k ′ 1···x k ′κ−1, R j′u i′ 1After linear combinations on the system (7.28) we obtain the following equations:⎧Π(x, u, J) = Rx j k1···x kκ,(7.28)Π(x, u, J) = Q l u i 1,Π(x, u, J) = R⎪⎨j u i 1u , i 2Π(x, u, J) = Q l x k1 x k2 x k3,Π(x, u, J) = R j x k1 u i 1 ,Π(x, u, J) = Q k 1x k1 x k2, k 1 ≠ k 2 ,⎪⎩ Π(x, u, J) = Q l x k1 x k2, l ≠ k 1 , l ≠ k 2 .Moreover all the partial derivatives (with respect to x l and u i ) up to orderthree of the coefficients Q l and R j of the vector field X ∈ Sym(E ) are ofthe form Π(x, u, J). Hence every function Q l and R j is uniquely determinedby the values at the origin of the n + n 2 + n + m Cn+κ−1 + m 2 partialderivatives (7.28). This implies that dim K Sym(E ) ≤ n 2 + 2n + m 2 +m Cn+κ−1.Proof. Since the second part of Lemma 8.1 is immediate let us establish onlythe identities (7.28). We first specify the indices in the equation (7.28) [3] asfollows: l = k κ = · · · = k 3 = k 2 = k 1 ; then l = k κ = · · · = k 3 = k 2 ≠k 1 ; and finally l = k κ = · · · = k 3 , k 3 ≠ k 2 , k 3 ≠ k 1 . This gives threeequations whose members on the right hand side are the same as those inthe equation (7.28) and whose members on the left hand side are the sameas those in the equation (7.28) [3] :(7.28) ⎧ ()Π⎪⎨ (Π⎪⎩ Πx,u,Q l′ ,Q l′x k ′1,R j′ ,R j′x k ′1,... ,R j′x k ′ 1···x k ′κ−1,R j′x,u,Q l′ ,Q l′x k ′1,R j′ ,R j′x k ′1,... ,R j′x k ′ 1···x k ′κ−1,R j′(x,u,Q l′ ,Q l′x k ′1,R j′ ,R j′x k ′1,... ,R j′x k ′ 1···x k ′κ−1,R j′u i′ 1u i′ 1u i′ 1))).= C 1 κ R j x k1 u i 1 − C2 κ δ j i 1Q k 1x k1 x k1,= R j x k1 u i 1 − C1 κ−1 δj i 1Q k 2x k1 x k2, k 2 ≠ k 1 ,= −δ j i 1Q k 3x k1 x k2, k 3 ≠ k 1 , k 3 ≠ k 2 .We remark that these three equations (after specialization of j = i 1 orof j ≠ i 1 and after some easy linear combinations) provide directly thefifth, sixth and seventh equations of (7.28). In particular we may replace

the values of the partial derivatives R j′x j ′1u i′ 1179and Q l′x k ′1x k ′2with k ′ 1 ≠ k′ 2 orl ′ ≠ k 1 ′ , l′ ≠ k 2 ′ appearing in the expressions Π of the second memberof (7.28) [1] by their values just obtained from the fifth, the sixth and theseventh equations of (7.28). This gives the first equation of (7.28).Then we specify the indices in (7.28) [2] as follows: l = k κ = · · · = k 3 =k 2 = k 1 ; then l = k κ = · · · = k 3 = k 2 ≠ k 1 ; then l = k κ = · · · = k 3 ,k 3 ≠ k 2 , k 3 ≠ k 1 ; and finally l = k κ = · · · = k 4 , l ≠ k 1 , l ≠ k 2 , l ≠ k 3 .This gives four equations, whose members on the right hand side are thesame as those in (7.28) and the members on the left hand side are the sameas those in (7.28) [2] :(7.28) ⎧⎪⎨⎪⎩ΠΠΠΠ((((x,u,Q l′ ,Q l′x k ′1,Q l′x k ′1x k ′2,R j′ ,R j′x k ′1,... ,R j′x k ′1 ···x k ′κ−1,R j′= C 2 κ R j x k1 x k1 u i 1 − C3 κ δ j i 1Q k 1x k1 x k1 x k1,x,u,Q l′ ,Q l′x k ′1,Q l′x k ′1x k ′2,R j′ ,R j′x k ′1,... ,R j′x k ′1 ···x k ′κ−1,R j′u i′ 1u i′ 1,R j′x k ′1u i′ 1,R j′x k ′1u i′ 1= C 1 κ−1 Rj x k1 x k2 u i 1 − C2 κ−1 δj i 1Q k 2x k1 x k2 x k2, k 2 ≠ k 1 ,x,u,Q l′ ,Q l′x k ′1,Q l′x k ′1x k ′2,R j′ ,R j′x k ′1,... ,R j′x k ′1 ···x k ′κ−1,R j′u i′ 1,R j′x k ′1u i′ 1= R j x k1 x k2 u − i 1 C1 κ−2 δ j i 1Q k 3x k1 x k2 x k3, k 3 ≠ k 1 , k 3 ≠ k 2 ,)x,u,Q l′ ,Q l′x k ′1,Q l′x k ′1x k ′2,R j′ ,R j′x k ′1,... ,R j′x k ′1 ···x k ′κ−1,R j′= −δ j i 1Q l x k1 x k2 x k3, l ≠ k 1 , l ≠ k 2 , l ≠ k 3 .u i′ 1,R j′x k ′1u i′ 1Using the fifth, the sixth and the seventh equations of (7.28) just obtained,we may replace the partial derivatives R j′and Q l′x k ′ x kwith k′ 1 ′ ≠ k′ 2 or1 2x j ′1u i′ 1l ′ ≠ k 1, ′ l ′ ≠ k 2 ′ appearing in the expressions Π of (7.28), providing four newequations in which the arguments of Π are the desired ones: (x, u, J), whereJ is defined in (7.28):(7.28) ⎧Π(x, u, J) = Cκ 2 R j − x k1 x k1 u i 1 C3 κ δ j i 1Q k 1⎪⎨x k1 x k1 x k1,Π(x, u, J) = C 1 κ−1 Rj x k1 x k2 u i 1 − C2 κ−1 δj i 1Q k 2x k1 x k2 x k2, k 2 ≠ k 1 ,Π(x, u, J) = R j − x k1 x k2 u i 1 C1 κ−2 δj i 1Q k 3x k1 x k2 x k3, k 3 ≠ k 1 , k 3 ≠ k 2 ,⎪⎩ Π(x, u, J) = − δ j i 1Q l x k1 x k2 x k3, l ≠ k 1 , l ≠ k 2 , l ≠ k 3 .)))====

180Let us differentiate now the equations (7.28) with respect to the variablesx l as follows: first we differentiate (7.28) 1 with respect to x k1 ; then wedifferentiate (7.28) 2 with respect to x k2 ; finally we differentiate (7.28) 3 withrespect to x k3 . The arguments in the expressions Π in the equation (7.28)contain now the terms Rx j′k ′ ···x 1 k; we replace them by their value given in′ κthe first equation of (7.28) already obtained. The arguments also contain theterms R j′and Q l′x k ′ x kwith k′ 1 ′ ≠ k′ 2 or l′ ≠ k 1 ′ , l′ ≠ k 2 ′ . We replace1 2x j ′1u i′ 1them by their value given by the fifth, the sixth and the seventh equationsof (7.28). We obtain three new equations in which the arguments of theexpressions Π are the desired ones: (x, u, J), where J is defined in (7.28):(7.28)⎧⎪⎨⎪⎩Π (x, u, J) = C 1 κ Rj x k1 x k1 u i 1 − C2 κ δj i 1Q k 1x k1 x k1 x k1,Π (x, u, J) = R j x k1 x k2 u i 1 − C1 κ−1 δj i 1Q k 2x k1 x k2 x k2, k 2 ≠ k 1 ,Π (x, u, J) = −δ j i 1Q k 3x k1 x k2 x k3, k 3 ≠ k 1 , k 3 ≠ k 2 .The seven equations (7.28) and (7.28) may be considered as three systemsof two linear equations of two variables with a nonzero determinant, theseventh equation being the last equation in (7.28). We immediately obtain:(7.28) ⎧Π(x, u, J) = R j = x k1 x k1 u i 1 δj i 1Q k 1x k1 x k1 x k1,⎪⎨Π(x, u, J) = R j x k1 x k2 u i 1 = δj i 1Q k 2x k1 x k2 x k2, k 2 ≠ k 1 ,Π(x, u, J) = R j = x k1 x k2 u i 1 δj i 1Q k 3x k1 x k2 x k3, k 3 ≠ k 1 , k 3 ≠ k 2 ,⎪⎩ Π(x, u, J) = δ j i 1Q l x k1 x k2 x k3, k 3 ≠ k 1 , k 3 ≠ k 2 ,giving the fourth equation in (7.28).It remains now to obtain the second and the third equations in (7.28).Let us write firstly equation (7.28) [6] with the choice of the indices j = i 1 ,l = k 1 = · · · = k κ . This gives the equation:(7.28)⎧ (Q ⎪⎨l u i 1= Π x, u, Q l′ , Q l′x k, . . .,Q l′′ x 1 k ′ 1···x k, R j′ , R′ x j′κ−2k, . . ., R′ x j′1k ′ ···x 1 k,′κ−1)⎪⎩, R j′, R j′, . . .,R j′u i′ 1 x k ′ u i′ 1 x 1 k ′ 1···x k ′ u i′ 1κ−3We observe first that the differentiation with respect to the variables x l ofone of the expressions Π(x, u, J) remains an expression Π(x, u, J). Indeedwe see from (7.28) that there appears, in the partial derivative J xl , derivativesQ l′x k ′ x 1 kwith k′ 1 ′ ≠ k′ 2 or l′ ≠ k 1 ′ , l′ ≠ k 2 ′ . We may replace them2by their value obtained in the sixth and the seventh equations of (7.28). It.

also appears some derivatives Q l′x k ′1x k ′2x k ′3(we replace them by their valueobtained in the fourth equation of (7.28)), some derivatives Rx j′k ′ ···x k(we′1 κreplace them by their value obtained in the first equation of (7.28)) andsome derivatives R j′(we replace them by their value obtained in thex k ′1u i′ 1fifth equation of (7.28)). Consequently we may write:181(7.28) [Π(x, u, J)] xl = Π(x, u, J).It follows that any derivative with respect to x l (to any order) of the fourthand the fifth equations of (7.28) provides expressions of the form Π(x, u, J).In other words for any integer λ ≥ 3 and any integer µ ≥ 1 we have(7.28)⎧⎨ Π(x, u, J) = Q l x k1 x k2 x k3 ···x kλ,⎩ Π(x, u, J) = R j . x k1 ···x kµ u i 1We may replace then these values in the equation (7.28), replacing also thederivatives Q l′x k ′ x 1 kwith k′ 1 ′ ≠ k′ 2 or l′ ≠ k 1 ′ , l′ ≠ k 2 ′ by their values obtainedin the sixth and the seventh equations of (7.28). This gives the second2equation of (7.28).We also remark that by a differentiation with respect to the variables x l ,the second equation Q l = Π(x, u, J) just obtained implies, using (7.28):u i 1(7.28) Π(x, u, J) = Q l x k1 u i 1.It remains finally to write (7.28) [4] first with the choice of indices l = k 1 =· · · = k κ , j = i 1 then with the choice of indices l = k 1 = · · · = k κ , j ≠ i 1 .We also write (7.28) [5] first with the choice of indices l = k 1 = · · · = k κ ,j = i 2 then with the choice of indices l = k 1 = · · · = k κ , j ≠ i 1 , j ≠ i 2 .We obtain four new equations:

182(7.28) ⎧(R i 1u i 1u − i 1 κQk 1x k1 u = Π x,u,Q l′ ,Q l′i 1 x k,... ,Q l′′ x 1 k ′ ···x 1 k,Q l′′κ−1 u i′ 1 ,Rj′ ,Rx j′k,... ,R j′′1⎪⎨⎪⎩,R j′,R j′,... ,R j′,u i′ 1 x k ′ u i′ 1 x k ′ ···x k ′ u i′ 11 1 κ−2(R j u i 1u = Π x,u,Q l′ ,Q l′i 1 x k,... ,Q l′′ x k ′ ···x k,Q l′′1 1 κ−1 u i′ 1 ,Rj′ ,Rx j′k,... ,R j′′1)x k ′1 ···x k ′κ−1,x k ′1 ···x k ′κ−1,,R j′,R j′,... ,R j′, j ≠ iu i′ 1 x k ′ u i′ 1 x k ′ ···x k ′ u i′ 1 ,11 1 κ−2(2R i 2u i 1u − i 2 κQk 1x k1 u = Π x,u,Q l′ ,Q l′i 1 x k,... ,Q l′′ x 1 k ′ ···x 1 k,Q l′′κ−1 u i′ 1 ,Rj′ ,Rx j′k,... ,R′ x j′1k ′ ···x 1 k,′κ−1),R j′,R j′,... ,R j′, iu i′ 1 x k ′ u i′ 1 x k ′ ···x k ′ u i′ 1 ≠ i 2 ,11 1 κ−2(R j u i 1u = Π x,u,Q l′ ,Q l′i 2 x k,... ,Q l′′ x k ′ ···x k,Q l′′1 1 κ−1 u i′ 1 ,Rj′ ,Rx j′k,... ,R′ x j′k ′ ···x k,′1 1 κ−1),R j′,R j′,...,R j′u i′ 1 x k ′ u i′ 1 x k ′ ···x k ′ u i′ 11 1 κ−2Using the equations of (7.28) we already obtained (namely all except thesecond equation), using (7.28) and (7.28), we may simplify these four equations:⎧Π(x, u, J) = R i 1u⎪⎨i 1u , i 1Π(x, u, J) = R j u(7.28)i 1u , j ≠ i i 1 1,Π(x, u, J) = R i 1u⎪⎩i 1u , i i 2 1 ≠ i 2 ,Π(x, u, J) = R j u i 1u , i i 2 1 ≠ i 2 , j ≠ i 1 , j ≠ i 2 .This gives the second equation of (7.28), completing the proof of Lemma 8.1and consequently the proof of Theorem 6.4.REFERENCES[1] BAOUENDI, M.S.; EBENFELT, P.; ROTHSCHILD, L.P.: Real submanifolds in complexspace and their mappings. Princeton Mathematical Series, 47, Princeton UniversityPress Princeton, NJ, 1999, xii+404 pp.[2] BLUMAN, G.W.; KUMEI, S.: Symmetries and differential equations, Springer Verlag,Berlin, 1989.[3] CARTAN, É.: Sur la géométrie pseudo-conforme des hypersurfaces de l’espace dedeux variables complexes, I, Annali di Mat. 11 (1932), 17–90.[4] CHERN, S.S.; MOSER, J.K.: Real hypersurfaces in complex manifolds, Acta Math.133 (1974), no.2, 219–271.), i 1 ≠ i 2 , j ≠ i 2 , j ≠ i 2 .

[5] F. ENGEL; LIE, S.: Theorie der Transformationsgruppen, I, II, II, Teubner, Leipzig,1889, 1891, 1893.[6] FELS, M.: The equivalence problem for systems of second-order ordinary differentialequations, Proc. London Math. Soc. 71 (1995), 221–240.[7] GAUSSIER, H.; MERKER, J.: A new example of uniformly Levi degenerate hypersurfacein C 3 , Ark. Mat., to appear.[8] GAUSSIER, H.; MERKER, J.: Nonalgebraizable real analytic tubes in C n , Math. Z.,to appear.[9] GAUSSIER, H.; MERKER, J.: Sur l’algébrisabilité locale de sous-variétés analytiquesréelles génériques de C n , C. R. Acad. Sci. Paris Sér. I Math., to appear.[10] GAUSSIER, H.; MERKER, J.: Géométrie des sous-variétés analytiques réelles de C net symétries de Lie des équations aux dérivées partielles, Bull. Soc. Math. Tunisie, toappear.[11] GONZÁLEZ-GASCÓN, F.; GONZÁLEZ-LÓPEZ, A.: Symmetries of differential equations,IV. J. Math. Phys. 24 (1983), 2006–2021.[12] GONZÁLEZ-LÓPEZ, A.: Symmetries of linear systems of second order differentialequations, J. Math. Phys. 29 (1988), 1097–1105.[13] IBRAGIMOV, N.H.: Group analysis of ordinary differential equations and the invarianceprinciple in mathematical physics, Russian Math. Surveys 47:4 (1992), 89–156.[14] LIE, S.: Theorie der Transformationsgruppen, Math. Ann. 16 (1880), 441–528.[15] MERKER, J.: Vector field construction of Segre sets, Preprint 1998, augmented in2000. Downloadable at arXiv.org/abs/math.CV/9901010.[16] MERKER, J.: On the partial algebraicity of holomorphic mappings between two realalgebraic sets, Bull. Soc. Math. France 129 (2001), no.3, 547–591.[17] MERKER, J.: On the local geometry of generic submanifolds of C n and the analyticreflection principle, Viniti, to appear.[18] OLVER, P.J.: Applications of Lie groups to differential equations. Springer Verlag,Heidelberg, 1986.[19] OLVER, P.J.: Equivalence, Invariance and Symmetries. Cambridge, Cambridge UniversityPress, 1995, xvi+525 pp.[20] POINCARÉ, H.: Les fonctions analytiques de deux variables et la représentation conforme,Rend. Circ. Mat. Palermo, II, Ser. 23, 185–220.[21] SEGRE, B.: Intorno al problema di Poincaré della rappresentazione pseudoconforme,Rend. Acc. Lincei, VI, Ser. 13 (1931), 676–683.[22] SEGRE, B.: Questioni geometriche legate colla teoria delle funzioni di due variabilicomplesse, Rendiconti del Seminario di Matematici di Roma, II, Ser. 7 (1932), no. 2,59–107.[23] STORMARK, O.: Lie’s structural approach to PDE systems. Encyclopædia of mathematicsand its applications, vol. 80, Cambridge University Press, Cambridge, 2000,xv+572 pp.[24] SUKHOV, A.: Segre varieties and Lie symmetries, Math. Z. 238 (2001), no.3, 483–492.[25] SUKHOV, A.: On transformations of analytic CR structures, Pub. Irma, Lille 2001,Vol. 56, no. II.[26] SUKHOV, A.: CR maps and point Lie transformations, Michigan Math. J. 50 (2002),369–379.[27] SUSSMANN, H.J.: Orbits of families of vector fields and integrability of distributions,Trans. Amer. Math. Soc. 180 (1973), 171-188.183

184[28] TRESSE, A.: Détermination des invariants ponctuels de l’équation différentielle dusecond ordre y ′′ = ω(x, y, y ′ ), Hirzel, Leipzig, 1896.

185Nonrigid sphericalreal analytic hypersurfaces in C 2Joël MerkerAbstract. A Levi nondegenerate real analytic hypersurface M of C 2 representedin local coordinates (z,w) ∈ C 2 by a complex defining equation of the formw = Θ(z,z,w) which satisfies an appropriate reality condition, is spherical if andonly if its complex graphing function Θ satisfies an explicitly written sixth-orderpolynomial complex partial differential equation. In the rigid case (known before),this system simplifies considerably, but in the general nonrigid case, its combinatorialcomplexity shows well why the two fundamental curvature tensors constructedby Élie Cartan in 1932 in his classification of hypersurfaces have, since then, neverbeen reached in parametric representation.arxiv.org/abs/0910.1694/Table of contents1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.2. Segre varieties and partial differential equations . . . . . . . . . . . . . . . . . . . . . . . . . 220.3. Geometry of associated submanifolds of solutions. . . . . . . . . . . . . . . . . . . . . . . . .224.4. Effective differential characterization of sphericality in C 2 . . . . . . . . . . . . . . . . 226.5. Some complete expansions: examples of expression swellings. . . . . . . . . . . . . .210.§1. INTRODUCTIONA real analytic hypersurface M in C 2 is called spherical at one of itspoints p if there exists a nonempty open neighborhood U p of p in C 2such that M ∩ U p is biholomorphic to a piece of the unit sphere S 3 ={(z, w): |z| 2 + |w| 2 = 1 } . When M is connected, sphericality at onepoint is known to propagate all over M, for it is equivalent to the vanishingof two certain real analytic curvature tensors that were constructed byÉlie Cartan in [3]. However, the intrinsic computational complexity, in theCauchy-Riemann (CR for short) context, of Élie Cartan’s algorithm to derivean absolute parallelism on some suitable eight-dimensional principalbundle P → M prevents from controlling explicitly all the appearing differentialforms. As a matter of fact, the effective computation, in terms of adefining equation for M, of the two fundamental differential invariants thevanishing of which characterizes sphericality, appears nowhere in the literature(see e.g. [23, 5, 11] and the references therein as well), except notablywhen one makes the assumption that, in some suitable local holomorphic

186coordinates (z, w) = (x + iy, u + iv) vanishing at the point p, the definingequation is of the so-called rigid form u = ϕ(x, y) with the variable vmissing, or even of the so-called (simpler) tube form u = ϕ(x), with thetwo variables y and v missing, see [11] which showed recently a renewedinterest, in CR geometry, for explicit characterizations of sphericality. Butin general, a real analytic hypersurface M ⊂ C 2 is represented at p by a realequation u = ϕ(x, y, v) whose graphing function ϕ depends entirely arbitrarilyupon v also, and then apparently, the characterization of sphericalityis still unknown.On the other hand, in the studies [12, 13, 14, Me2005a, Me2005b] devotedto the CR reflection principle, it was emphasized that all the adequateinvariants of CR mappings between CR manifolds: Pair of Segre foliations,Segre chains, Complexified CR orbits, Jets of complexified Segre varietes,Rigidity of formal CR mappings, Nondegeneracy conditions, CR-reflectionfunction 5 , can be viewed correctly only when M is represented by a socalledcomplex defining equation of the form:w = Θ ( z, z, w ) ,where the function Θ ∈ C { z, z, w } , vanishing at the origin, is the uniquew+wfunction obtained by solving with respect to w the equation: =2ϕ ( z+z, z−z,) w−w2 2i 2i ; then the fact that ϕ was real is reflected, in terms of thisnew function Θ(z, z, w), by the constraint that, together with its complexconjugate Θ ( z, z, w ) , it satisfies the functional equation 6 :w ≡ Θ ( z, z, Θ(z, z, w) ) .Accordingly, the author suspected since a few years — cf. the Open Question2.35 in [19] — that sphericality of M at p should and could be expressedadequately in terms of Θ. The classical assumption that M be Levinondegenerate at the point p (see e.g. [11]) — which is the origin of ourpresent system of coordinates (z, w) — may then be expressed here (cf.[Me2005a, Me2005b]) by requiring that Θ z Θ zw −Θ w Θ zz does not vanish atthe origin. In particular, this guarantees that the following explicit rationalexpression whose numerator is a polynomial in the fourth-order jet J 4 z,z,w Θ,is well defined and analytic in some sufficiently small neighborhood of the5 For a presentation of these concepts, the reader is referred to the extensive introductionsof [14, Me2005b] and also to [19] for more about why dealing only with complexdefining equations is natural and unavoidable when one wants to insert CR geometry in thewider universe of completely integrable systems of real or complex analytic partial differentialequations.6 More will be said shortly in Section 2 below.

origin:{ ( ∣ ∣ )AJ 4 1∣∣∣ Θ(Θ) :=[Θ z Θ zw − Θ w Θ zz ] 3 Θ zzzz Θ w Θ z Θ w ∣∣∣w −Θ zz Θ zw( ∣ ∣ ) ( ∣ ∣)∣∣∣ Θ− 2Θ zzzw Θ z Θ z Θ w ∣∣∣ ∣∣∣ Θw + ΘΘ zz Θ zzww Θ z Θ z Θ w ∣∣∣z +zw Θ zz Θ zw( ∣ ∣ ∣ ∣ ∣ ∣ )∣∣∣ Θ+ Θ zzz Θ z Θ w Θ ww ∣∣∣ ∣∣∣ Θz − 2ΘΘ zw Θ z Θ w Θ zw ∣∣∣ ∣∣∣ Θw + Θzww Θ zw Θ w Θ w Θ zz ∣∣∣w +zzw Θ zw Θ zzz( ∣ ∣ ∣ ∣ ∣ ∣)}∣∣∣ Θ+ Θ zzw − Θ z Θ z Θ ww ∣∣∣ ∣∣∣ Θz + 2ΘΘ zz Θ z Θ z Θ zw ∣∣∣ ∣∣∣ Θw − Θzww Θ zz Θ w Θ z Θ zz ∣∣∣w .zzw Θ zz Θ zzzWe hope, then, that the following precise statement will fill a gap in ourunderstanding of the vanishing of CR curvature tensors.Main (and unique) theorem. An arbitrary, not necessarily rigid, real analytichypersurface M ⊂ C 2 which is Levi nondegenerate at one of its pointsp and has a complex defining equation of the form:w = Θ ( z, z, w )in some system of local holomorphic coordinates (z, w) ∈ C 2 centered at p,is spherical at p if and only if its graphing complex function Θ satisfies thefollowing explicit sixth-order algebraic partial differential equation:(0 ≡− Θ w ∂Θ z Θ zw − Θ w Θ zz ∂z +identically in C { z, z, w } .Θ zΘ z Θ zw − Θ w Θ zz∂∂w) 2[AJ 4 (Θ) ]Here, it is understood that the first-order derivation in parentheses is appliedtwice to the fourth-order rational differential expression AJ 4 (Θ). The1factor[Θ z Θ zw −Θ w Θ zzthen appears, and after clearing out this denominator,] 7one obtains a universal polynomial differential expression AJ 6 (Θ) dependingupon the sixth-order jet Jz,z,w 6 Θ and having integer coefficients. A partialexpansion is provided in Section 5, and the already formidable incompressiblelength of this expansion perhaps explains the reason why no referencein the literature provides the explicit expressions, in terms of some definingfunction for M, of Élie Cartan’s two fundamental differential invariants 7which can (in principle) be used to classify real analytic hypersurfaces ofC 2 up to biholomorphisms, and to at least characterize sphericality.Suppose in particular for instance that M is rigid, given by a complexequation of the form w = −w + Ξ(z, z), that is to say with Θ(z, z, w)of the form −w + Ξ(z, z), so that the reality condition simply reads here:Ξ(z, z) ≡ Ξ(z, z). Then as a corollary-exercise, sphericality is explicitly7 See [3] and also [23], where the tight analogy with second-order ordinary differentialequations is well explained.187

188characterized by a much simpler partial differential equation that we canwrite down in expanded form:0 ≡ Ξ z 2 z 4(Ξzz) 4− 6 Ξ z 2 z 3 Ξ zz 2(Ξzz) 5− 4 Ξ z 2 z 2 Ξ zz 3(Ξzz) 5− Ξ z 2 z Ξ zz 4(Ξzz) 5++ 15 Ξ ( ) 2z 2 z 2 Ξzz 2( ) 6+ 10 Ξ zz 3 Ξ z 2 z Ξ zz 2( ) 6− 15 Ξ ( 3z 2 z Ξzz 2)( ) 7,Ξzz Ξzz Ξzzand this equation should of course hold identically in C { z, z } .Now, here is a summarized description of our arguments of proof. BeniaminoSegre ([23]) in 1931 and in fact much earlier Sophus Lie himself inthe 1880’s (see e.g. Chapter 10 of Volume I of the Theorie der Transformationsgruppen[8]) showed how to elementarily associate a unique secondorderordinary differential equation:w zz (z) = Φ ( z, w(z), w z (z) )to the Levi nondegenerate equation w = Θ(z, z, w) by eliminating the twovariables z and w, viewed as parameters, from the two equations w = Θand w z = Θ z . We check in great details the semi-known result that M isspherical at the origin if and only if its associated differential equation isequivalent, under some appropriate local holomorphic point transformation(z, w) ↦−→ (z ′ , w ′ ) = ( z ′ (z, w), w ′ (z, w) ) fixing the origin, to the simplestpossible equation w z ′ ′ z ′(z′ ) = 0 having null right-hand side, whose obvioussolutions are just the affine complex lines. But since the doctoral dissertationof Arthur Tresse (defended in 1895 under the direction of Lie in Leipzig),it is known that, attached to any such differential equation are two explicitdifferential invariants:I 1 := Φ wzw zw zw zand:I 2 := DD ( Φ wzw z)− Φwz D ( Φ wzw z)− 4 D(Φwwz)++ 6 Φ ww − 3 Φ w Φ wzw z+ 4 Φ wz Φ wwz ,where D :=∂ z + w z ∂ w + Φ(z, w, w z ) ∂ wz ,depending both upon the fourth-order jet of Φ, which, together with all theircovariant differentiations, enable one (in principle 8 ) to completely determinewhen two arbitrarily given differential equations are equivalent one toanother 9 . A very well-known application is: the vanishing of both I 1 and8 To our knowledge, the only existing reference where this strategy is seriously endeavouredin order to classify second-order ordinary differential equations y xx (x) =F ( x, y(x), y x (x) ) is [HK1989], but only for certain point transformations — called there“fiber-preserwing” — of the special form (x, y) ↦→ (x ′ , y ′ ) = ( x ′ (x), y ′ (x, y) ) , the firstcomponent of which is independent of y.9 Three decades earlier, Christoffel in his famous memoir [4] of 1869 devoted to theequivalence problem for Riemannian metrics discovered that the covariant differentiations

I 2 characterizes equivalence to w ′ z ′ z ′(z′ ) = 0. So in order to characterizesphericality, one only has to reexpress the vanishing of I 1 and of I 2 in termsof the complex defining function Θ(z, z, w). For this, we apply the techniquesof computational differential algebra developed in [19] which enableus here to explicitly execute the two-ways transfer between algebraic expressionsin the jet of Φ and algebraic expressions in the jet of Θ. It thenturns out that the two equations which one obtains by transferring to Θ thevanishing of I 1 and of I 2 are conjugate one to another, so that a single equationsuffices, and it is precisely the one enunciated in the theorem. In fact,this coincidence is caused by the famous projective duality, explained e.g.by Lie and Scheffers in Chapter 10 of [12] and restituted in modern languagein [1, 5]. It is indeed well known that to any second-order ordinarydifferential equation (E ): y xx (x) = F ( x, y(x), y x (x) ) is canonically associateda certain dual second-order ordinary differential equation, call it (E ∗ ):b aa (a) = F ∗( a, b a (a), b aa (a) ) , which has the crucial property that:I 1 (E ) is a nonzero multiple of I 2 (E ∗ )and symmetrically also: I 2 (E ) is a nonzero multiple of I 1 (E ∗ ).The doctoral dissertation [10] of Koppisch (Leipzig 1905) cited only passimby Élie Cartan in [Ca1924] contains the analytical details of this correspondence,which was well reconstituted recently in [5] within the context ofprojective Cartan connections. But the differential equation which is dual tothe one w zz (z) = Φ ( z, w(z), w z (z) ) associated to w = Θ(z, z, w) is easilyseen to be just its complex conjugate (E ): w zz (z) = Φ ( )z, w(z), w z , andthen as a consequence, I 1 = (E) I1 (E ) is the conjugate of I1 (E), and similarly alsoI 2 = (E) I2 (E ) is the conjugate of I2 (E ). So it is no mystery that, as said, thesphericality of M at the origin:0 ≡ I 1 (E) and 0 ≡ I 2 (E ) = nonzero · I 1 (E ) = nonzero · I1 (E ) ,can in a simpler way be characterized by the vanishing of the two mutuallyconjugate (complex) equations:0 ≡ I 1 (E) and 0 ≡ I 1 (E ) ,which of course amount to just one (complex) equation.To conclude this introduction, we would like to mention firstly that noneof our computations — especially those of Sections 4 and 5 — was performedwith the help of any computer, and secondly that the effective characterizationof sphericality in higher complex dimension n 3 will appearsoon [21].of the curvature provide a full list of differential invariants for positive definite quadraticinfinitesimal metrics.189

190§2. SEGRE VARIETIES AND DIFFERENTIAL EQUATIONSReal analytic hypersurfaces in C 2 . Let us consider an arbitrary real analytichypersurface M in C 2 and let us localize it around one of its points, sayp ∈ M. Then there exist complex affine coordinates:(z, w) = ( x + iy, u + iv )vanishing at p in which T p M = {u = 0}, so that M is represented in aneighborhood of p by a graphed defining equation of the form:where the real-valued function:ϕ = ϕ(x, y, v) =∑u = ϕ(x, y, v),k,l,m∈Nk+l+m2ϕ k,l,m x k y l v m ∈ R { x, y, u } ,which possesses entirely arbitrary real coefficients ϕ k,l,m , vanishes at theorigin: ϕ(0) = 0, together with all its first order derivatives: 0 = ∂ x ϕ(0) =∂ y ϕ(0) = ∂ v ϕ(0). All studies in the analytic reflection principle 10 showwithout doubt that the adequate geometric concepts: Pair of Segre foliations,Segre chains, Complexified CR orbits, Jets of complexified Segre varietes,Rigidity of formal CR mappings, Nondegeneracy conditions, CR-reflectionfunction, can be viewed correctly only when M is represented by a so-calledcomplex defining equation. Such an equation may be constructed by simplyrewriting the initial real equation of M as:w+w= ϕ ( z+z, z−z,w−w2 2 2i 2iand then by solving 11 the so written equation with respect to w, which yieldsan equation of the shape 12 :w = Θ ( z, z, w ) =∑Θ α,β,γ z α z β w γ ∈ C { z, z, w } ,α, β, γ ∈ Nα+β+γ1whose right-hand side converges of course near the origin (0, 0, 0) ∈ C ×C × C and has complex coefficients Θ α,β,γ ∈ C. The paradox that anysuch complex equation provides in fact two real defining equations for thereal hypersurface M which is one-codimensional, and also in addition thefact that one could as well have chosen to solve the above equation withrespect to w, instead of w, these two apparent “contradictions” are corrected10 The reader might for instance consult the survey [18], pp. 5–44 or the memoirs[Me2005a, Me2005b], and look also at some of the concerned references therein.11 Thanks to dϕ(0) = 0, the holomorphic implicit function theorem readily applies.12 Notice that since dϕ(0) = 0, one has Θ = −w + order 2 terms.),

y means of a fundamental, elementary statement that transfers to Θ (in anatural way) the condition of reality:ϕ(x, y, u) =∑ϕ k,l,m x k y l v m =∑ϕ k,l,m x k y l v m = ϕ(x, y, v)k+l+m1k+l+m1enjoyed by the initial definining function ϕ.Theorem. ([18], p. 19 13 ) The complex analytic function Θ = Θ(z, z, w)with Θ = −w + O(2) together with its complex conjugate 14 :Θ = Θ ( z, z, w) =∑Θ α,β,γ z α z β w γ ∈ C { z, z, w }α, β, γ∈Nsatisfy the two (equivalent by conjugation) functional equations:(7.28)w ≡ Θ ( z, z, Θ(z, z, w) ) ,w ≡ Θ ( z, z, Θ(z, z, w) ) .Conversely, given a local holomorphic function Θ(z, z, w) ∈ C{z, z, w},Θ = −w + O(2) which, in conjunction with its conjugate Θ(z, z, w), satisfiesthis pair of equivalent identities, then the two zero-sets:{ ( )} { ( )}0 = −w + Θ z, z, w and 0 = −w + Θ z, z, wcoincide and define a local one-codimensional real analytic hypersurfaceM passing through the origin in C 2 .As before, let M be an arbitrary real analytic hypersurface passingthrough the origin in C 2 equipped with coordinates (z, w), and assume thatT 0 M = {u = 0}. Without loss of generality, we can and we shall assumethat the coordinates are chosen in such a way that a certain standard convenientnormalization condition holds.Theorem. ([Me2005a], p. 12) There exists a local complex analytic changeof holomorphic coordinates h: (z, w) ↦−→ (z ′ , w ′ ) = h(z, w) fixing theorigin and tangent to the identity at the origin of the specific form:z ′ = z,w ′ = g(z, w),such that the image M ′ := h(M) has a new complex defining equationw ′ = Θ ′( z ′ , z ′ , w ′) satisfying:Θ ′( 0, z ′ , w ′) ≡ Θ ′( z ′ , 0, w ′) ≡ −w ′ ,13 Compared to [18], we denote here by Θ the function denoted there by Θ.14 According to a general, common convention, given a power series Φ(t) =∑γ∈N n Φ γ t γ , t ∈ C n , Φ γ ∈ C, one defines the series Φ(t) := ∑ γ∈N n Φ γ t γ by conjugatingonly its complex coefficients. Then the complex conjugation operator distributesoneself simultaneously on functions and on variables: Φ(t) ≡ Φ(¯t), a trivial property whichis nonetheless frequently used in the formal CR reflection principle ([Me2005a, Me2005b]).191

192or equivalently, which has a power series expansion of the form:Θ ′( z ′ ,z ′ ,w ′) = − w ′ +∑∑ ∑Θ ′ α,β,0 z′ α z′β + w ′ γΘ ′ α,β,γ z′ α z′β .α1, β1γ1α1, β1Levi nondegenerate hypersurfaces. Leaving aside the real defining equationof M, let us now rename the complex defining equation of M in suchnormalized coordinates simply as before: w = Θ(z, z, w), dropping all theprime signs. Quite concretely, the real analytic hypersurface M is said to beLevi nondegenerate at the origin if the coefficient Θ 1,1,0 of zz, which may bechecked to always be real because of the reality condition (7.28), is nonzero.In fact, it is well known that Levi nondegeneracy is a biholomorphically invariantproperty, see for instance [18], p. 158, but in more conceptual terms,the following general characterization, which may be taken as a definitionhere, holds true. One then readily checks that it is equivalent to Θ 1,1,0 ≠ 0in normalized coordinates.Lemma. ([Me2005a, Me2005b, 19]) The real analytic hypersurface M ⊂C 2 with 0 ∈ M represented in coordinates (z, w) by a complex definingequation of the form w = Θ(z, z, w) is Levi nondegenerate at the origin ifand only if the map:(z,w)↦−→(Θ(0, z, w), Θz (0, z, w) )has nonvanishing 2 × 2 Jacobian determinant at (z, w) = (0, 0).After a possible real dilation of the z-coordinate, we can therefore assumethat Θ 1,1,0 = 1, and then we are provided with the following normalization:(7.28) w = −w + zz + zz O ( |z| + |w| ) ,that will be useful shortly. Another, even more convincing argument forconsigning to oblivion the real defining equation u = ϕ(x, y, v) dates backto Beniamino Segre [23], who observed that to any real analytic M are associatedtwo deeply linked objects.1) The nowadays so-called Segre varieties 15 S q associated to any pointq ∈ C 2 near the origin of coordinates (z q , w q ) that are the complexcurves defined by the equation:S q := { 0 = −w + Θ ( z, z q , w q)},quite appropriately in terms of the fundamental complex defining functionΘ; this equation is holomorphic just because its antiholomorphicterms are set fixed.15 A presentation of the general theory, valuable for generic CR manifolds of arbitrarycodimension d 1 and of arbitrary CR dimension m 1 in C m+d enjoying no specificnondegeneracy condition, may be found in [Me2005a, Me2005b, 18].

2) When M is Levi nondegenerate at the origin, a second-order complexordinary differential equation 16 of the form:w zz (z) = Φ ( z, w(z), w z (z) ) ,193whose solutions are exactly the Segre varieties of M, parametrized bythe two initial conditions w(0) and w z (0) which correspond bijectivelyto the antiholomorphic variables z q and w q .In fact, the recipe for deriving the second-order differential equation associatedto a local Levi-nondegenerate M ⊂ C 2 with 0 ∈ M represented bya normalized 17 equation of the form (7.28) is very simple. Considering thatw = w(z) is given in the equation:w(z) = Θ ( z, z, w )as a function of z with two supplementary (antiholomorphic) parameters zand w that one would like to eliminate, we solve with respect to z and w,just by means of the implicit function theorem 18 , the pair of equations:[w(z) = Θ(z, z, w)= −w + zz + zz O(|z| + |w|)w z (z) = Θ z(z, z, w)= z + z O(|z| + |w|)the second one being obtained by differentiating the first one with respect toz, and this yields a representation:z = ζ ( z, w(z), w z (z) ) and w = ξ ( z, w(z), w z (z) )for certain two uniquely defined local complex analytic functionsζ(z, w, w z ) and ξ(z, w, w z ) of three complex variables. By means ofthese functions, we may then replace z and w in the second derivative:( )w zz (z) = Θ zz z, z, w= Θ zz(z, ζ(z, w(z), wz (z) ) , ξ ( z, w(z), w z (z) ))=: Φ ( z, w(z), w z (z) ) ,and this defines without ambiguity the associated differential equation.More about differential equations will be said in §3 below.16 This idea, usually attributed by contemporary CR geometers to B. Segre, dates in factback (at least) to Chapter 10 of Volume 1 of the 2 100 pages long Theorie der Transformationsgruppenwritten by Sophus Lie and Friedrich Engel between 1884 and 1893, where itis even presented in the uppermost general context.17 In fact, such a normalization was made in advance just in order to make things concreteand clear, but thanks to what the Lemma on p. 192 expresses in a biholomorphicallyinvariant way, everything which follows next holds in an arbitrary system of coordinates.18 Justification: by our preliminary normalization, the 2 × 2 Jacobian determinant∂(Θ, Θ z)∂(z, w)computed at the origin equals˛ 0 −1˛1 0 ˛, hence is nonzero. Without the preliminarynormalization, the condition of the Lemma on p. 192 also applies in any case.

194Of course, any spherical real analytic M ⊂ C 2 must be Levi nondegenerateat every point, for the unit 3-sphere S 3 ⊂ C 2 is. It is well known thatS 3 minus one of its points, for instance: S 3 \ {p ∞ } with p ∞ := (0, −1), isbiholomorphic, through the so-called Cayley transform:(z, w) ↦−→ ( iz1+w , 1−w2+2 wto the so-called Heisenberg sphere of equation:)=: (z ′ , w ′ ) having inverse: (z ′ , w ′ ) ↦−→ ( −2iz ′1+2w ′ ,1−2w ′1+2 w ′ )w ′ = −w ′ + z ′ z ′ ,in the target coordinate-space (z ′ , w ′ ), and this model will be more convenientto deal with for our purposes.Proposition. A Levi nondegenerate local real analytic hypersurface M inC 2 is locally biholomorphic to a piece of the Heisenberg sphere (hencespherical) if and only if its associated second-order ordinary complex differentialequation is locally equivalent to the Newtonian free particle equation:w ′ z ′ z ′(z′ ) = 0, with identically vanishing right-hand side.Proof. Indeed, any local equivalence of M to the Heisenberg sphere transformsits differential equation to the one associated with the Heisenbergsphere, and then trivially: w ′ z ′(z′ ) = z ′ , whence w ′ z ′ z ′(z′ ) = 0.Conversely, if the Segre varieties of M are mapped to the solutions ofw ′ z ′ z ′(z′ ) = 0, namely to the complex affine lines of C 2 , the complex definingequation of the transformed M ′ must necessarily be affine:(7.28) w ′ = λ ′ (z ′ , w ′) + z ′ µ ′( z ′ , w ′) =: Θ ′( z ′ , z ′ , w ′) ,with certain coefficients that are holomorphic with respect to (z ′ , w ′ ). Thenλ ′ (0) = 0 since the origin is fixed, and if µ ′ (0) is nonzero, one performsthe linear transformation z ′ ↦→ z ′ , w ′ ↦→ w ′ − µ ′ (0) z ′ , which stabilizes bothw ′ z ′ z ′(z′ ) = 0 and the form of (7.28), to insure then that µ ′ (0) = 0.Next, the second reality condition (7.28) now reads:w ′ ≡ λ ′ (z ′ , Θ ′ (z ′ , z ′ , w ′)) + z ′ µ ′( z ′ , Θ ′ (z ′ , z ′ , w ′)) ,and by differentiating it with respect to z ′ , we get, without writing the argumentsfor brevity:0 ≡ λ ′ z + ′ Θ′ z ′ λ′ w + ′ z′ µ ′ z + ′ z′ Θ ′ z ′ µ′ w ′≡ λ ′ z + ′ µ′ λ ′ w + ′ z′ µ ′ z + ′ z′ µ ′ µ ′ w ′,where we replace Θ ′ z in the second line by its value ′µ′ (z ′ , w ′ ). But withall the arguments, this identity reads in full length as the following identity

195holding in C { z ′ , z ′ , w ′} :− λ ′ z ′ (z ′ , Θ ′ (z ′ , z ′ , w ′)) − z ′ µ ′ z ′ (z ′ , Θ ′ (z ′ , z ′ , w ′)) ≡≡ µ ′ (z ′ , w ′ ) λ ′ w ′ (z ′ , Θ ′ (z ′ , z ′ , w ′)) + z ′ µ ′ (z ′ , w ′ ) µ ′ w ′ (z ′ , Θ ′ (z ′ , z ′ , w ′)) .For convenience, it is better to take (z ′ , z ′ , w ′ ) as arguments of this identityinstead of (z ′ , z ′ , w ′ ), so we simply replace w ′ in it by:Θ ′( z ′ , z ′ , w ′) ,we apply the first reality condition (7.28) and we get what we wanted topursue the reasonings:(7.28)−λ ′ (z z ′ , w ′) (− z ′ µ ′ ′ z z ′ , w ′) ≡ µ ′( z ′ , λ ′ (z ′ , w ′ ) + z ′ µ ′ (z ′ , w ′ ) )·′[· λ ′ (w z ′ , w ′) (+ z ′ µ ′ ′ w z ′ , w ′)] ,′i.e. an identity holding now in C { z ′ , z ′ , w ′} . The left-hand side being affinewith respect to z ′ , the same must be true of each one of the two factors of theright-hand side. In particular, the second order derivative of the first factorwith respect to z ′ must vanish identically:{0 ≡ ∂ z ′∂ ( z ′ µ′z ′ , λ ′ + z ′ µ ′)}≡ µ ′ z ′ z ′ + 2 µ′ µ ′ z ′ w ′ + µ′ µ ′ µ ′ w ′ w ′.Because M ′ is Levi nondegenerate at the origin, the lemma on p. 192 togetherwith the affine form (7.28) of the defining equation entails that themap:((7.28) z ′ , w ′) ↦−→ ( λ ′ (z ′ , w ′ ), µ ′ (z ′ , w ′ ) )has nonvanishing Jacobian determinant at (z ′ , w ′ ) = (0, 0). Consequently,in the above identity (rewritten with some of the arguments):0 ≡ µ ′ z ′ z ′ (z ′ , λ ′ +z ′ µ ′) +2 µ ′ µ ′ z ′ w ′ (z ′ , λ ′ +z ′ µ ′) +µ ′ µ ′ µ ′ w ′ w ′ (z ′ , λ ′ +z ′ µ ′) ,we can consider z ′ , λ ′ and µ ′ as being just three independent variables. Settingµ ′ = 0, we get 0 ≡ µ ′ z ′ z ′ (z ′ , λ ′ ), that is to say: µz ′ z ′(z′ , w ′ ) ≡ 0 andthen after division of µ ′ , we are left with only two terms:0 ≡ 2 µ ′ z ′ w ′ (z ′ , λ ′ + z ′ µ ′) + µ ′ µ ′ w ′ w ′ (z ′ , λ ′ + z ′ µ ′) .Then again 0 ≡ 2 µ z ′ w ′(z′ , w ′ ) and finally also 0 ≡ µ w ′ w ′(z′ , w ′ ). This meansthat the function:µ ′ (z ′ , w ′ ) = c ′ 1 z′ + c ′ 2 w′ ,with some two constants c ′ 1 , c′ 2 ∈ C, is linear.

196Now, we claim that c ′ 2 = 0 in fact. Indeed, setting z ′ = 0 in (7.28), weget:−λ ′ z ′ (0, w′ ) −z ′ c ′ 1 ≡ { c ′ 1 z ′ +c ′ 2(λ′(0, w ′ )+z ′ c ′ 2w ′)} ·[λ ′ w ′(0, w′ )+z ′ c ′ 2].The coefficient c ′ 2c ′ 2c ′ 2 of (z ′ ) 2 w ′ in the right-hand side must vanish, so c ′ 2 =0. Since the rank at the origin of the map (7.28) equals 2, necessarily µ ′ ≢ 0,so c ′ 1 ≠ 0, and then c′ 1 = 1 after a suitable dilation of the z′ -axis. Next,rewriting the identity (7.28):−λ ′ z ′ (z ′ , w ′) − z ′ ≡ z ′[ λ ′ w ′ (z ′ , w ′)] ,we finally get λ ′ z ≡ 0 and ′ λ′ w ′ ≡ −1, which means in conclusion that:λ ′ (z ′ , w ′ ) ≡ −w ′ and µ ′ (z ′ , w ′ ) ≡ z ′ ,so that the equation of M ′ is the one: w ′ = −w ′ + z ′ z ′ of the Heisenbergsphere in the target coordinates (z ′ , w ′ ).Thanks to this proposition, in order to characterize the sphericality of alocal real analytic hypersurface M ⊂ C 2 explicitly in terms of its complexdefining function Θ, our strategy 19 will be to:□ characterize the local equivalence to w ′ z ′ z ′(z′ ) = 0 of the associateddifferential equation:(7.28) w zz (z) = Θ zz(z, ζ(z, w(z), wz (z) ) , ξ ( z, w(z), w z (z) )) ,explicitly in terms of the three functions Θ zz , ζ and ξ;□ eliminate any occurence of the two auxiliary functions ζ and ξ so asto re-express the obtained result only in terms of the sixth-order jet J 6 z,z,w Θ.§3. GEOMETRY OF ASSOCIATED SUBMANIFOLDS OF SOLUTIONSThe characterization we will obtain holds in fact inside a broader contextthan just CR geometry, in terms of what we called in [19] the submanifold ofsolutions associated to any second-order ordinary differential equation, nomatter whether it comes or not from a Levi nondegenerate M ⊂ C 2 . In fact,the elementary foundations towards a general theory embracing all systemsof completely integrable partial differential equations was laid down [19],especially by producing explicit prolongation formulas for infinitesimal Liesymmetries, with many interesting problems that are still wide open as soonas the number of (independent or dependent) variables increases: constructionof Cartan connections; production of differential invariants; full classificationaccording to the Lie symmetry group.19 — indicated already as the accessible Open Question 2.35 in [19] —

Fortunately for our present purposes here, the geometry, the classification,and the Lie transformation group features of second order ordinary differentialequations are essentially completely understood since the groundbreakingworks of Lie [Lie1883], followed by a prized thesis by Tresse [Tr1896]and later by a celebrated memoir of Élie Cartan, see also [GTW1989] andthe references therein.Accordingly, letting x ∈ K and y ∈ K be two real or complex variables(with hence K = R or C throughout), consider any second-order ordinarydifferential equation:y xx (x) = F ( x, y(x), y x (x) )having local K-analytic right-hand side F , and denote it by (E ) for short. Inthe space of first-order jets of arbitrary graphing functions y = y(x) that weequip with three independent coordinates denoted (x, y, y x ), let us introducethe vector field:D := ∂∂x + y ∂x∂y + F(x, y, y x) ∂ ,∂y xwhose integral curves inside the three-dimensional space (x, y, y x ) correspond,classically, to solving the equation y xx (x) = F(x, y(x), y x (x)) bytransforming it into a system of two first-order differential equations withthe two unknown functions y(x) and y x (x).Theorem. ([Lie1883, Tr1896, Ca1924, GTW1989, 17]) A second-order ordinarydifferential equation y xx = F(x, y, y x ) denoted (E ) with K-analyticright-hand side possesses two fundamental differential invariants, namely:I 1 (E) := F y xy xy xy xand:I 2 (E) := DD( F yxy x)− Fyx D ( F yxy x)− 4 D(Fyyx)++ 6 F yy − 3 F y F yxy x+ 4 F yx F yyx ,while all other differential invariants are deduced from I 1 (E ) and I2 (E)by covariant(in the sense of Tresse) or coframe (in the sense of Cartan) diffentiations.Moreover, local equivalence to y x ′ ′ x ′(x′ ) = 0 holds under someinvertible local K-analytic point transformation:(x, y) ↦−→ (x ′ , y ′ ) = ( x ′ (x, y), y ′ (x, y) )if and only if both invariants vanish:0 = I 1 (E ) = I 2 (E).In order to characterize sphericality of an M ⊂ C 2 , it is then natural andadvisable to study what the vanishing of the above two differential invariantsgives when applied to the second order ordinary differential equation (7.28)enjoyed by the defining function Θ. This goal will be pursued in §4 below.197

198For the time being, with the aim of extending such a kind of characterizationto a broader scope, following §2 of [19], let us now recall how one mayin a natural way construct a sumanifold of solutions M E associated to thedifferential equation (E ) which, when (E ) comes from a Levi nondegeneratelocal real analytic hypersurface M ⊂ C 2 , regives without any modificationits complex defining equation w = Θ ( z, z, w).To begin with, in the first-order jet space (x, y, y x ) that we simply drawas a common three-dimensional space:y xyD0abxyDM (E)0a, bxDDDexp(xD)(0,a, b)we duplicate the two dependent coordinates (y, y x ) by introducing a newsubspace of coordinates (a, b) ∈ K × K, and we draw a vertical plane containingthe two new axes that are just parallel copies (for the moment, justlook at the left-hand side). Then the leaves of the local foliation associatedto the integral curves of the vector field D are uniquely determined by theirintersection with this plane, because thanks to the presence of ∂ in D, all∂xthese curves are approximately directed by the x-axis in a neighborhood ofthe origin: no tangent vector can be vertical. But we claim that all such intersectionpoints of coordinates (0, b, a) ∈ K×K×K correspond bijectivelyto the two initial conditions y(0) ≡ b and y x (0) = a for solving uniquely thedifferential equation. In fact, the flow of D at time x starting from all suchpoints (0, b, a) of the duplicated vertical plane:exp(x D)(0, b, a) =: ( x, Q(x, a, b), S(x, a, b) )(see again the diagram) expresses itself in terms of two certain local K-analytic functions Q and S that satisfy, by the very definition of the flow ofour vector field ∂ x + y x ∂ y + F ∂ yx , the following two differential equations:ddx Q(x, a, b) = S(x, a, b) and: ddx S(x, a, b) = F( x, Q(x, a, b), S(x, a, b) )together with the (obious) initial condition for x = 0:(0, b, a) = exp(0 D)(0, b, a) = ( 0, Q(0, a, b), S(0, a, b) ) .

We notice passim that S ≡ Q x (no two symbols were in fact needed), andmost importantly, we emphasize that in this way, we have viewed in a somewhatgeometric-minded way of thinking that the general solution:y = y(x) = Q ( x, y x (0), y(0) ) = Q(x, a, b)to the original differential equation arises naturally as the first (amongst two)graphing function for the integral curves of D in the first order jet space,these curves being parametrized by (a, b).Definition. The sumanifold of solutions 20 M (E) associated with the secondorderordinary differential equation (E ): y xx (x) = F ( x, y(x), y x (x) ) isthe local K-analytic submanifold of the four-dimensional Euclidean spaceK x × K y × K a × K b represented as the zero-set:0 = −y + Q(x, a, b),where Q(x, a, b) is the general local K-analytic solution of (E ), satisfyingtherefore:Q xx(x, a, b) ≡ F(x, Q(x, a, b), Qx (x, a, b) ) ,and Q(0, a, b) = b, Q x (0, a, b) = a.Conversely, let us assume we are given a submanifold M of K x × K y ×K a × K b of the specific equation y = Q(x, a, b), for a certain local K-analytic function Q of the three variables (x, a, b). Call (x, y) the variables,(a, b) the parameters, and call M solvable with respect to the parameters (atthe origin) if the map:(a, b) ↦−→ ( Q(0, a, b), Q x (0, a, b) )has rank two at the central point (a, b) = (0, 0). Of course, the submanifoldof solutions associated to any second-order ordinary differential equationis solvable with respect to parameters, for in this case Q(0, a, b) ≡ b andQ x (0, a, b) ≡ a.Similarly as what we did for deriving 2) on p. 193, if an arbitrarily givensubmanifold M of K x × K y × K a × K b is assumed to be solvable withrespect to parameters, then viewing y in y = Q(x, a, b) as a parametrizedfunction of x, the implicit function theorem enables one to solve (a, b) in thetwo equations:[ y(x) = Q(x, a, b)y x (x) = Q x (x, a, b),19920 At this point, the reader is referred to [19] for more about how one can develope thewhole theory of Lie symmetries of partial differential equations intrinsically within submanifoldsof solutions only; the theory of Cartan connections associated to certain exteriordifferential systems could (and should also) be transferred to submanifolds of solutions.

200to yield both a representation for a and and a representation for b of the form:[ (a = A x, y(x), yx (x) )(7.28)b = B ( x, y(x), y x (x) ) ,for certain two local K-analytic functions A and B of three independentvariables (x, y, y x ), that one may insert afterwards in the second order derivative:y xx (x) = Q xx(x, a, b)= Q xx(x, A(x, y(x), yx (x)), B(x, y(x), y x (x)) )=: F ( x, y(x), y x (x) ) ,which yields the differential equation (E M ) associated to the submanifoldM solvable with respect to the parameters. In summary:Proposition. ([19]) There is a one-to-one correspondence:(E M ) = (E ) ←→ M = M (E )between second-order ordinary differential equations (E ) of the generalform:y xx (x) = F ( x, y(x), y x (x) )and submanifolds (of solutions) M of equation:y = Q(x, a, b)that are solvable with respect to the parameters, and this correspondencesatisfies:(EM(E))= (E ) and M(EM ) = M.We now claim that solvability with respect to the parameters is an invariantcondition, independently of the choice of coordinates. Indeed, lety = Q(x, a, b) be any submanifold of solutions, call it M , and let:(x, y, a, b)↦−→(x ′ (x, y), y ′ (x, y), a, b )be an arbitrary local K-analytic diffeomorphism fixing the origin( which leaves untouched the parameters. The vector of coordinates1, Qx (x, a, b), 0, 0 ) based at the point ( x, Q(x, a, b), a, b ) of M is sent,through such a diffeomorphism, to a vector whose x ′ -coordinate equals:ddx[x ′ (x, Q) ] = x ′ x +Q x x ′ y . Therefore the implicit function theorem insuresthat, provided the expression:x ′ x(x, y) + Q x (x, a, b) x ′ y(x, y) ≠ 0does not vanish, the image M ′ of M through such a diffeomorphism canstill be represented, locally in a neighborhood of the origin, as a graph of asimilar form:y ′ = Q ′( x ′ , a, b ) ,

for a certain local K-analytic new function Q ′ = Q ′ (x ′ , a, b). Since M : y =Q(x, a, b) is sent to M ′ : y ′ = Q ′ (x ′ , a, b), it follows that x ′ (x, y), y ′ (x, y),Q(x, a, b) and Q ′ (x ′ , a, b) are all linked by the following fundamental identity:(7.28) y ′( x, Q(x, a, b) ) ≡ Q ′( x ′ (x, Q(x, a, b)), a, b ) ,which holds in C { x, a, b } .Claim. If M is solvable with respect to the parameters (at the origin), thenM ′ is also solvable with respect to the parameters (at the origin too), andconversely.Proof. The assumption that M is solvable with respect to the parameters isequivalent to the fact that its first order x-jet map:(x, a, b)↦−→(x, Q(x, a, b), Qx (x, a, b) ))is (locally) of rank three. One should therefore look at the same first order jetmap attached to M ′ , represented in the right part of the following diagram:((x, a, b)x ′ (x, Q(x, a, b)), a, b )(x ′ , a, b),(x, Q(x, a, b), Qx (x, a, b) ) X ? ( x ′ , Q ′ (x ′ , a, b), Q ′ x ′(x′ , a, b) ) ( x ′ , Q ′ (x ′ , a, b), Q ′ x ′(x′ , a, b) )and ask how these two x- and x ′ -jet maps can be related to each other,namely search for a map:( ) (X ?: x, Q, Qx ↦−→ x, Q ′ , Q x)′which would close up the diagram and make it commutative.The answer for the second component of the sought map is simply:( )X 2 :( x, Q, Qx ↦−→ y′x, Q ) ,since (9) indeed shows that composing the right vertical arrow with the upperhorizontal one gives the same result, concerning a second component, ascomposing the bottom horizontal arrow with the left vertical one.The answer for the third component of the sought map then proceeds bydifferentiating with respect to x the fundamental identity (9), which yields,without writing the arguments:y ′ x + Q x y ′ y ≡ [ x ′ x + Q x x ′ y]Q′x ′,and since x ′ x + Q x x ′ y ≠ 0 by assumption, it suffices to set:X 3 :( ) y x ′ (x, Q) + Q x y y ′ (x, Q)x, Q, Qx ↦−→x ′ x (x, Q) + Q x x ′ y (x, Q),201

202in order to complete the commutativity of the diagram, namely to get:( ) yQ ′ x ′ (x, Q(x, a, b)) + Q x(x, a, b) y y ′ (x, Q(x, a, b))x f(x, Q(x, a, b)), a, b ≡ ′ x ′ x (x, Q(x, a, b)) + Q x(x, a, b) x ′ y (x, Q(x, a, b)),as was required. But now considering instead the inverse diffeomorphismechanges nothing to the reasonings, hence we have at the same time a rightinverse:((x, a, b)x ′ (x, Q(x, a, b)), a, b ) (x, a, b)x-jet(x, Q(x, a, b), Qx (x, a, b) ) X ( x ′ , Q ′ (x ′ , a, b), Q ′ x ′(x′ , a, b) ) X −1 ( ) x, Q(x, a, b), Q(x, a, b)x ′ -jetx-jetof our commutative diagram, so that the x-jet map and the x ′ -jet map havecoinciding ranks at pairs of points which correspond one to another.We are now in a position to generalize the characterization of sphericalityderived earlier on p. 223.Proposition. A second-order ordinary differential equation y xx (x) =F(x, y(x), y x (x)) with K-analytic right-hand side is equivalent, under someinvertible local K-analytic point transformation (x, y) ↦→ (x ′ , y ′ ), to thefree particle Newtonian equation y ′ x ′ x ′(x′ ) = 0 if and only if its associatedsubmanifold of solutions y = Q(x, a, b) is equivalent, under some local K-analytic map in which variables are separated from parameters:(x, y, a, b) ↦−→ ( x ′ (x, y), y ′ (x, y), a ′ (a, b), b ′ (a, b) )to the affine submanifold of solutions of equation y ′ = b ′ + x ′ a ′ .Before proceeding to the proof, let us observe that when one looks at areal analytic hypersurface M ⊂ C 2 , the corresponding transformation inthe parameter space is constrained to be the conjugate transformation of thelocal biholomorphism:(z, w, z, w)↦−→(z ′ (z, w), w ′ (z, w), z ′ (z, w), w ′ (z, w) ) ,while one has more freedom for general differential equations, in the sensethat transformations of variables and transformations of parameters are entirelydecoupled.Proof. One direction is clear: if y = Q(x, a, b) is equivalent to:(7.28) y ′ = b ′ + x ′ a ′ = b ′ (a, b) + x ′ a ′ (a, b),then its associated differential equation y xx (x) = F ( x, y(x), y x (x) ) isequivalent, through the same diffeomorphism (x, y) ↦→ (x ′ , y ′ ) of the variables,to the differential equation associated with (7.28), which trivially is:y ′ x ′ x ′(x′ ) = 0.

Conversely, if y xx (x) = F ( x, y(x), y x (x) ) is equivalent, through a diffeomorphism(x, y) ↦→ (x ′ , y ′ ), to yx ′ ′ x ′(x′ ) = 0, then its submanifoldof solutions y = Q(x, a, b) is transformed to y ′ = Q ′ (x ′ , a, b) and sincey x ′ ′ x ′(x′ ) = 0, the function Q ′ is necessarily of the form:y ′ = b ′ (a, b) + x ′ a ′ (a, b).Because the condition of solvability with respect to the parameters is invariant,the rank of (a, b) ↦→ ( a ′ (a, b), b ′ (a, b) ) is again equal to 2, whichconcludes the proof.Coming now back to the wanted characterization of sphericality, our moregeneral goal now amounts to characterize, directly in terms of its fundamentalsolution function Q(x, a, b), the local equivalence to y ′ x ′ x ′(x′ ) = 0 of asecond-order ordinary differential equation y xx (x) = F ( x, y(x), y x (x) ) .Afterwards at the end, it will suffice to replace Q(x, a, b) simply byΘ(z, z, w) in the obtained equations.But before going further, let us explain how a certain generalized projectiveduality will simplify our task, as already said in the Introduction. Thus,let (E ): y xx (x) = F ( x, y(x), y x (x) ) be a differential equation as abovehaving general solution y = Q(x, a, b) = −b+xa+O(x 2 ), with initial conditionsb = −y(0) and a = y x (0). The implicit function theorem enables usto solve b in the equation y = Q(x, a, b) of the associated submanifold ofsolutions M (E) in terms of the other quantities, which yields an equation ofthe shape:b = Q ∗ (a, x, y) = −y + ax + O(x 2 ),for some new local K-analytic function Q ∗ = Q ∗ (a, x, y). Then similarly aspreviously, we may eliminate x and y from the two equations:b(a) = Q ∗ (a, x, y) = −y + ax + O(x 2 )b a (a) = Q ∗ a (a, x, y) = x + O(x2 ),that is to say: x = X ( a, b(a), b a (a) ) and y = Y ( a, b(a), b a (a) ) , and we theninsert these two solutions in:b aa (a) = Q ∗ aa (a, x, y)(= Q ∗ aa a, X(a, b(a), ba (a)), Y (a, b(a), b a (a)) )=: F ∗( a, b(a), b a (a) ) .We shall call the so obtained second-order ordinary differential equation thedual of y xx (x) = F ( x, y(x), y x (x) ) .In the case of a hypersurface M ⊂ C 2 , solving w in the equation w =Θ(z, z, w) gives nothing else but the conjugate equation w = Θ(z, z, w),203

204just by virtue of the reality identities (7.28). It also follows rather triviallythat the dual differential equation:w zz (z) = Θ zz(z, ζ(z, w(z), wz (z)), ξ(z, w(z), w z (z)) )= Φ ( z, w(z), w z (z) )is also just the conjugate differential equation.To the differential equation y xx = F and to its dual b aa = F ∗ are associatedtwo submanifolds of solutions:M = M (E ) := {( x, y, a, b ) ∈ K × K × K × K: y = Q(x, a, b) } ,together with:M ∗ = M (E ∗ ) := {( a, b, x, y ) ∈ K × K × K × K: b = Q ∗ (a, x, y) } ,and as one obviously guesses, the duality, when viewed within submanifoldsof solutions, just amounts to permute variables and parameters:M ∋ (x, y, a, b) ←→ (a, b, x, y) ∈ M ∗ .In the CR case, if we denote by ˜z and ˜w two independent complex variableswhich correspond to the complexifications of z and w (respectively ofcourse), the duality takes place between the so-called extrinsic complexification([13, 14, Me2005a, Me2005b, 18, 19]):M = M c := {( z, w, ˜z, ˜w ) ∈ C × C × C × C: w = Θ ( z, ˜z, ˜w )}of M in one hand, and in the other hand, its own transformation 21 :M ∗ = ∗ c (M c ) := {(˜z, ˜w, z, w ) ∈ C × C × C × C: ˜w = Θ (˜z, z, w )}under the involution:∗ c( z, w, ˜z, ˜w ) := (˜z, ˜w, z, w )which clearly is the complexification of the natural antiholomorphic involution:∗ ( z, w, z, w ) := ( z, w, z, w )that fixes M pointwise, as it fixes any other real analytic subset of C 2 . Here,one has M ∗ = ∗(M ) — which is ≠ M in general — and of course also(M∗ )∗ = M .So in terms of the coordinates (x, a, b) on M and of the coordinates(a, x, y) on M ∗ , the duality is the map:(x, a, b) ↦−→ ( a, x, Q(x, a, b) )with inverse:(a, x, y) ↦−→ ( x, a, Q ∗ (a, x, y) ) .21 Be careful not to write {( z, w, ˜z, ˜w ) : ˜w = Θ (˜z, z, w )} , because this would regivethe same subset M of C 2 × C 2 , due to the reality identities (7.28).

205But we may also express the duality from the first jet (x, y, y x )-space to thefirst jet (a, b, b a )-space by simply composing the following three maps, thecentral one being the duality M → M ∗ :⎛⎝⎞(a, x, y) (( ↓a, Q ∗ (a, x, y), Q ∗ a(a, x, y) ) ⎠ ◦⎛(x, a, b) → ( a, x, Q(x, a, b) )) ◦ ⎝(x, A(x, y, yx ), B(x, y, y x ) )↑(x, y, y x )which in sum gives us the map:(A(x, y, yx ), Q(x, y, y x ) ↦−→∗( A(x, y, y x ), x, Q ( x, A(x, y, y x ), B(x, y, y x ) )) )( ,A(x, y, yx ), x, Q ( x, A(x, y, y x ), B(x, y, y x ) )) .Q ∗ aWith the approximations, one checks that:(x, y, y x ) ↦−→ ( y x + · · · , −y + xy x + · · · , x + · · ·),where the remainder terms “+ · · · ” are all O(x 2 ). For the differential equationy xx (x) = 0 of affine lines, these remainders disappear completely andwe recover the classical projective duality written in inhomogeneous coordinates([5], pp. 156–157). Furthermore, one shows (see e.g. [5]) that theabove duality map within first order jet spaces is a contact transformation,namely through it, the pullback of the standard contact form db−b a da in thetarget space is a nonzero multiple of the standard contact form dy − y x dx inthe source space.But what matters more for us is the following. The two fundamentaldifferential invariants of b aa (a) = F ∗( a, b(a), b a (a) ) are functions exactlysimilar to the ones written on p. 197, namely:⎞⎠,I 1 (E ∗ ) := F ∗ b ab ab ab aI 2 (E ∗ ) := D ∗ D ∗( F ∗ b ab a)− F∗baD ∗( F ∗ b ab a)− 4 D∗ ( F ∗ bb a)++ 6 F ∗ bb − 3 F ∗ b F ∗ b ab a+ 4 F ∗ b aF ∗ bb a,where D ∗ := ∂ a + b a ∂ b + F ∗ (a, b, b a ) ∂ ba . Then according to Koppisch([10]), through the duality map, I 1 (E)is transformed to a nonzero multiple ofI 2 (E ∗ ) , and simultaneously also, I2 (E ) is transformed to a nonzero multiple22 ofI 1 (E ∗ ) , so that: 0 = I 1 (E) ⇐⇒ I 2 (E ∗ ) = 00 = I 2 (E) ⇐⇒ I 1 (E ∗ ) = 0.Consequently, the differential equation (E ): y xx (x) = F ( x, y(x), y x (x) ) isequivalent to y ′ x ′ x ′(x′ ) = 0 if and only if:F yxy xy xy x= 0 and F ∗ b ab ab ab a= 0 .22 To be precise, both factors of multiplicity ([5], p. 165) are nonvanishing in a neighboroodof the origin, but for our purposes, it suffices just that they are not identically zeropower series.

206This observation has essentially no practical interest, because the computationof F ∗ in terms of F relies upon the composition of three maps . . .except notably in the CR case, since the duality in this case is complex conjugation:Φ ∗ = Φ. In summary, we have established the following.Proposition. An arbitrary, not necessarily rigid, real analytic hypersurfaceM ⊂ C 2 which is Levi nondegenerate at one of its points p and has a complexdefinining equation of the form:w = Θ ( z, z, w )in some system of local holomorphic coordinates (z, w) ∈ C 2 centered at p,is spherical at p if and only if the right-hand side Φ of its uniquely associatedsecond-order ordinary complex differential equation:w zz (z) = Φ ( z, w(z), w z (w) )satisfies the single fourth-order partial differential equation:0 ≡ Φ wzw zw zw z(z, w, wz).It now only remains to re-express this fourth-order partial differentialequation in terms of the complex graphing function Θ(z, z, w) for M. Wewill achieve this more generally for F yxy xy xy x.§4. EFFECTIVE DIFFERENTIAL CHARACTERIZATIONOF SPHERICALITY IN C 2Reminding the reasonings and notations introduced in a neighborhood ofequation (7.28), the transformation:(x, y, yx)↦−→(x, a, b)and its inverse are given by the two triples of functions:⎡⎡x = xx = x⎢⎣ a = A(x, y, y x )b = B(x, y, y x )and⎢⎣ y = Q(x, a, b)y x = Q x (x, a, b).Equivalently, one has the two pairs of identically satisfied equations:a ≡ A ( x, Q(x,a,b), Q x (x,a,b) )b ≡ B ( x, Q(x,a,b), Q x (x,a,b) ) and y ≡ Q ( x, A(x,y,y x ), B(x,y,y x ) )y x ≡ Q x(x, A(x,y,yx ), B(x,y,y x ) ) .Differentiating the second column of equations with respect to x, to y and toy x yields:0 = Q x + Q a A x + Q b B x 0 = Q xx + Q xa A x + Q xb B x1 = Q a A y + Q b B y 0 = Q xa A y + Q xb B y0 = Q a A yx + Q b B yx 1 = Q xa A yx + Q xb B yx .

Then thanks to a straightforward application of the rule of Cramer for 2 × 2linear systems, we derive six useful formulas.Lemma. ([19], p. 9) All the six first order derivatives A x , A y , A yx , B x ,B y , B yx of the two functions A and B with respect to their three arguments(x, y, y x ) may be expressed as follows in terms of the second jet J 2 (Q) ofthe defining function Q:A x = Q b Q xx − Q x Q xbQ a Q xb − Q b Q xa,Q xbA y =,Q a Q xb − Q b Q xaB y =−Q bA yx =,Q a Q xb − Q b Q xaB yx =B x = Q x Q xa − Q a Q xx,Q a Q xb − Q b Q xa−Q xa,Q a Q xb − Q b Q xaQ aQ a Q xb − Q b Q xa.For future abbreviation, we shall denote the single appearing denominator— which evidently is the common determinant of all the three 2 × 2linear systems involved above — simply by a square symbol:∆ := Q a Q xb − Q b Q xa .The two-ways transfer between functions G defined in the (x, y, y x )-spaceand functions T defined in the (x, a, b)-space, namely the one-to-one correspondence:G(x, y, y x ) ←→ T(x, a, b)may be read very concretely as the following two equivalent identities:G(x, y, y x ) ≡ T ( x, A(x, y, y x ), B(x, y, y x ) )G ( x, Q(x, a, b), Q x (x, a, b) ) ≡ T(x, a, b),holding in K{x, y, y x } and in K{x, a, b} respectively. By differentiatingthe first identity, the chain rule shows how the three first-order derivationoperators (basic vector fields) ∂ x , ∂ y and ∂ yx living in the (x, y, y x )-spaceare transformed into the (x, a, b)-space:∂∂x = ∂ ( ) ( )∂x + Qb Q xx − Q x Q xb ∂∆ ∂a + Qx Q xa − Q a Q xx ∂∆ ∂b( ) (Qxb ∂∂a + −Qxa∂∂y =∂∂y x=) ∂∆∆ ∂b( ) ( ) −Qb ∂∆ ∂a + Qa ∂∆ ∂b .Lemma. The total differentiation operator D = ∂ x +y x ∂ y +F ∂ yx associatedto y xx = F(x, y, y x ) simply transfers to the basic derivation operator alongthe x-direction:D ←→ ∂ x .207

208Proof. Reading the three formulas just preceding, by adding the first one tothe second one multiplied by y x = Q x together with the third one multipliedby F = Q xx , one visibly sees that the coefficients of both ∂ ∂a and ∂ ∂b dovanish in the obtained sum, as announced.Keeping in mind — so as to avoid any confusion — that the same letter xis used to denote simultaneously the independent variable of the differentialequation y xx = F(x, y, y x ) and the non-parameter variable of the associatedsubmanifold of solutions y = Q(x, a, b), we may now write this two-waystransfer D ←→ ∂ x exactly as we did in the above three equations, namelysimply as an equality between two derivations living in the (x, y, y x )-spaceand in the (x, a, b)-space:D = ∂ x .Lemma. With G = G(x, y, y x ) being any local K-analytic function in the(x, y, y x )-space, the three second-order derivatives G yxy x, G yyx and G yy expressas follows in terms of the second-order jet Jx,a,b 2 (T) of the definingfunction T :G yxy x= Q b Q b∆ 2+ T a∆ 3 (+ T b∆ 3 (G yyx = − Q b Q xb∆ 2+ T (a∆ 3+ T b∆ 3 (G yy = Q xb Q xb∆ 2+ T (a∆ 3+ T b∆ 3 (T aa − 2 Q a Q b∆ 2 T ab + Q a Q a∆ 2 T bb +∣ ∣ ∣ ∣ ∣ ∣)∣∣∣ QQ a Q b Q bb ∣∣∣ ∣∣∣ Qa − 2 QQ xb Q a Q b Q ab ∣∣∣ ∣∣∣ Qb + Qxbb Q xb Q b Q b Q aa ∣∣∣b +xab Q xb Q xaa∣ ∣ ∣ ∣ ∣ ∣)∣∣∣ Q− Q a Q a Q bb ∣∣∣ ∣∣∣ Qa + 2 QQ xa Q a Q a Q ab ∣∣∣ ∣∣∣ Qb − Qxbb Q xa Q b Q a Q aa ∣∣∣bxab Q xa Q xaaT aa + Q a Q xb + Q b Q xa∆ 2T ab − Q a Q xa∆ 2 T bb +∣ ∣ ∣∣∣ Q− Q a Q b Q bb ∣∣∣xa + ( ) ∣ ∣ ∣ ∣)QQ xb Q a Q xb + Q b Q xa ∣∣ Q b Q ab ∣∣∣ ∣∣∣ Q− Qxbb Q xb Q b Q b Q aa ∣∣∣xb +xab Q xb Q xaa∣ ∣ ∣∣∣ QQ a Q a Q bb ∣∣∣xa − ( ) ∣ ∣ ∣ ∣)QQ xa Q a Q xb + Q b Q xa ∣∣ Q a Q ab ∣∣∣ ∣∣∣ Q+ Qxbb Q xa Q b Q a Q aa ∣∣∣xbxab Q xa Q xaaT aa − 2 Q xa Q xb∆ 2T ab + Q xa Q xa∆ 2 T bb +∣ ∣ ∣ ∣ ∣ ∣)∣∣∣ QQ xa Q b Q bb ∣∣∣ ∣∣∣ Qxa − 2 QQ xb Q xa Q b Q ab ∣∣∣ ∣∣∣ Qxb + Qxbb Q xb Q xb Q b Q aa ∣∣∣xb +xab Q xb Q xaa∣ ∣ ∣ ∣ ∣ ∣)∣∣∣ Q− Q xa Q a Q bb ∣∣∣ ∣∣∣ Qxa + 2 QQ xa Q xa Q a Q ab ∣∣∣ ∣∣∣ Qxb − Qxbb Q xa Q xb Q a Q aa ∣∣∣xb .xab Q xa Q xaaProof. We apply the operatororder derivative G yx , namely we consider:∂∂y x, wiewed in the (x, a, b)-space, to the first∂ yx(Gyx)=∂∂y x[− Q b∆ T a + Q a∆ T b],

and we then expand carefully the result by collecting somewhat in advancethe obtained terms with respect to the derivatives of T :(G yxy x= − Q b ∂∆ ∂a + Q )[a ∂− Q b∆ ∂b ∆ T a + Q ]a∆ T b(Qb Q ab∆ − Q )b Q b ∆ a∆ ∆ 2 T a + Q b Q b∆ ∆ T aa+=∆(+ − Q b∆(+ − Q a∆(Qa+∆Q aa∆ + Q )b Q a ∆ a∆ ∆ 2 T b − Q b∆Q bb∆ + Q )a Q b ∆ b∆ ∆ 2 T a − Q a∆Q ab∆ − Q a∆)Q a ∆ b∆ 2 T b + Q a∆Q a∆ T ab+Q b∆ T ab+Q a∆ T bb.The terms involving T aa , T ab , T bb are exactly the ones exhibited by the lemmafor the expression of G yxy x. In the four large parentheses which are coefficientsof T a , T b , T a , T b , we replace the occurences of ∆ a , ∆ a , ∆ b , ∆ b simplyby:∆ a = Q xb Q aa + Q a Q xab − Q xa Q ab − Q b Q xaa∆ b = Q xb Q ab + Q a Q xbb − Q xa Q bb − Q b Q xab ,and the total sum of terms coefficiented by T a in our expression now becomes:T(a [ ] [ ]∆ 3 Q b Q ab Qa Q xb − Q b Q xa − Qb Q b Qxb Q aa + Q a Q xab − Q xa Q ab − Q b Q xaa −[ ] [ ] )− Q a Q bb Qa Q xb − Q b Q xa + Qa Q b Qxb Q ab + Q a Q xbb − Q xa Q bb − Q b Q xab == T (a∆ 3 Q a Q b Q xb Q ab − Q b Q b Q xa Q ab1 − Q b Q b Q xb Q aa − Q a Q b Q b Q xab ++ Q b Q b Q xa Q ab1 + Q b Q b Q b Q xaa −− Q a Q a Q xb Q bb + Q a Q b Q xa Q bb2 + Q a Q b Q xb Q ab + Q a Q a Q b Q xbb −)− Q a Q b Q xa Q bb − Q 2a Q b Q b Q xab == T (a [ ] [ ]∆ 3 Q a Q a Qb Q xbb − Q xb Q bb − 2 Qa Q b Qb Q xab − Q xb Q ab ++ Q b Q b[Qb Q xaa − Q xb Q aa] ) ,so that we now have effectively reconstituted the three 2 × 2 determinantsappearing in the second line of the expression claimed by the lemma forthe transfer of G yxy xto the (x, a, b)-space. The treatment of the coefficientof T bmakes only a few differences, hence will be skipped here (but not∆ 3in the manuscript). Finally, the two remaining expressions for G yyx andfor G yy are obtained by performing entirely analogous algebrico-differentialcomputations.End of the proof of the Main Theorem. Applying the above formula forG yxy xwith x := z, with a := z, with b := w, with ∆ := Θ z Θ zw − Θ w Θ zz ,209

210with G := Φ and with T := Θ zz , we exactly get the expression [ ] AJ 4 (Θ) ofthe Introduction, and then its further derivative ∂ yx ∂ yx Gyxyx = Gyxyxy xy xis exactly:(− Θ w0 ≡Θ z Θ zw − Θ w Θ zz=:AJ 6 (Θ)[Θ z Θ zw − Θ w Θ zz ] 7 .∂∂z +Θ zΘ z Θ zw − Θ w Θ zz) 2∂ [AJ 4 (Θ) ]∂wAs we have said, the vanishing of the second invariant of w zz (z) =Φ ( z, w(z), w z (z) ) amounts to the complex conjugation of the above equation,which is then obviously redundant. Thus, the proof of the Main Theoremis now complete, but we will nevertheless discuss in a specific finalsection what AJ 6 (Θ) would look like in purely expanded form.§5. SOME COMPLETE EXPANSIONS:EXAMPLES OF EXPRESSION SWELLINGSComing back to the non-CR context with the submanifold of solutionsM (E) = { y = Q(x, a, b) } , let us therefore figure out how to expand theexpression differentiated twice:G yxy xy xy x =„− Q b∆+ Ta∆ 3 „+ T b∆ 3 „∂∂a + Qa∆« 2 j∂ Qb Q bT∂b ∆ 2 aa − 2 Qa Q b Qa QaT∆ 2 ab + T∆ 2 bb +Q bbQQ a Q bQa˛˛˛˛ Q xb Q xbb˛˛˛˛ − 2 Q a Q b Q ab Qb˛˛˛˛ Q xb Q xab˛˛˛˛ + Q b Q b Q aab˛˛˛˛ Q xaa˛˛˛˛Q bbQ xbQ aQ a Q− Q a Q a˛˛˛˛ Q xa Q xbb˛˛˛˛ + 2 Q a Q ab Q a Q aab˛˛˛˛ Q xa Q xab˛˛˛˛ − Q b Q b˛˛˛˛ Q xaa˛˛˛˛which would make the Main Theorem a bit more precise and explicit.First of all, we notice that, in the formulas for G yxy x, for G yyx , for G yy ,all the appearing 2 × 2 determinants happen to be modifications of the basicJacobian-like ∆-determinant:∆ ( a|b ) := ∆ =∣ Q ∣a Q b ∣∣∣,Q xa Q xband we will denote them accordingly by employing the following (formallyand intuitively clear) notations:∆ ( b|bb ) :=∣ Q ∣b Q bb ∣∣∣∆ ( b|ab ) :=Q xb Q xbb∣ Q ∣b Q ab ∣∣∣∆ ( b|aa ) :=Q xb Q xab∣ Q ∣b Q aa ∣∣∣Q xb Q xaa∆ ( a|bb ) :=∣ Q ∣a Q bb ∣∣∣∆ ( a|ab ) :=Q xa Q xbb∣ Q ∣a Q ab ∣∣∣∆ ( a|aa ) :=Q xa Q xab∣ Q ∣a Q aa ∣∣∣,Q xa Q xaathe bottom line always coinciding with the differentiation with respect tox of the top line. These abbreviations will be very appropriate for the nextQ xa«+«ff,

explicit computation, so let us rewrite the formula for G yxy xusing this newlyintroduced formalism:{1 [G yxy x=∆(a|b) 3 T aa Qb Q b ∆(a|b) ] [+ T ab − 2Qa Q b ∆(a|b) ] [+ T bb Qa Q a ∆(a|b) ] +211+ T a[Qa Q a ∆(b|bb) − 2Q a Q b ∆(b|ab) + Q b Q b ∆(b|aa) ] ++ T b[− Qa Q a ∆(a|bb) + 2Q a Q b ∆(a|ab) − Q b Q b ∆(a|aa) ]} .Then the twelve partial derivatives with respect to a and with respect to b ofall the six determinants ∆ ( ∗|∗ ) appearing in the the second line are easy towrite down:[ ( ) ( )∂∂b ∆ b|bb ] = ∆ bb|bb + ∆( b|bbb ) [ ( ) ( ) ( )∂0 ∂a ∆ b|bb ] = ∆ ab|bb + ∆ b|abb∂∂b∂∂b∂∂b∂∂b∂∂b[∆(b|ab)] = ∆(bb|ab)+ ∆(b|abb)[∆(b|aa)] = ∆(bb|aa)+ ∆(b|aab)[∆(a|bb)] = ∆(ab|bb)+ ∆(a|bbb)∂∂a∂∂a∂∂a[∆(b|ab)] = ∆(ab|ab)0 + ∆( b|aab )[∆(b|aa)] = ∆(ab|aa)+ ∆(b|aaa)[∆(a|bb)] = ∆(aa|bb)+ ∆(a|abb)[ ( ) ( )∆ a|ab ] = ∆ ab|ab + ∆( a|abb ) [ ( ) ( ) ( )∂0 ∂a ∆ a|ab ] = ∆ aa|ab + ∆ a|aab[ ( ) ( ) ( ) [ ( ) ( )∆ a|aa ] = ∆ ab|aa + ∆ a|aab∂∂a ∆ a|aa ] = ∆ aa|aa + ∆( a|aaa ) ,0and the underlined terms vanish for the trivial reason that any 2 ×2 determinant,two columns of which coincide, vanishes. Consequently, we may nowendeavour the computation of the third order derivative:(G yxy xy x=− Q b∆∂∂a + Q a∆∂∂bWhen applying the two derivations in parentheses to:G yxy x= 1 ∆ 3 {expression}) [Gyxyx].we start out by differentiating1 multiplied by expression, and then we∆ 3differentiate expression. Before any contraction, the full expansion of:∆ 5 G yxy xy x=(we indeed clear out the denominator ∆ 5 ) is then:= T aaˆ3Qb Q b Q b ∆(a|b)∆(aa|b) + 3Q b Q b Q b ∆(a|b)∆(a|ab) − 3Q aQ b Q b ∆(a|b)∆(ab|b) − 3Q aQ b Q b ∆(a|b)∆(a|bb)˜++ T abˆ− 6QaQ b Q b ∆(a|b)∆(aa|b) − 6Q aQ b Q b ∆(a|b)∆(a|ab) + 6Q aQ aQ b ∆(a|b)∆(ab|a) + 6Q aQ aQ b ∆(a|b)∆(a|bb)˜++ T bbˆ3QaQ aQ b ∆(a|b)∆(aa|b) + 3Q aQ aQ b ∆(a|b)∆(a|ab) − 3Q aQ aQ a∆(a|b)∆(ab|b) − 3Q aQ aQ a∆(a|b)∆(a|bb)˜++T ah3Q aQ aQ b ∆(b|bb)∆(aa|b) + 3Q aQ aQ b ∆(b|bb)∆(a|ab) − 3Q aQ aQ a∆(b|bb)∆(ab|b) − 3Q aQ aQ a∆(b|bb)∆(a|bb)−− 6Q aQ b Q b ∆(b|ab)∆(aa|b) − 6Q aQ b Q b ∆(b|ab)∆(a|ab) + 6Q aQ aQ b ∆(b|ab)∆(ab|b) + 6Q aQ aQ b ∆(b|ab)∆(a|bb)+i+ 3Q b Q b Q b ∆(b|aa)∆(aa|b) + 3Q b Q b Q b ∆(b|aa)∆(a|ab) − 3Q aQ b Q b ∆(b|aa)∆(ab|b) − 3Q aQ b Q b ∆(b|aa)∆(a|bb) +

212+T bh3Q aQ aQ b ∆(a|bb)∆(aa|b) + 3Q aQ aQ b ∆(a|bb)∆(a|ab) − 3Q aQ aQ a∆(a|bb)∆(ab|b) − 3Q aQ aQ a∆(a|bb)∆(a|bb)−− 6Q aQ b Q b ∆(a|ab)∆(aa|b) − 6Q aQ b Q b ∆(a|ab)∆(a|ab) + 6Q aQ aQ b ∆(a|ab)∆(ab|b) + 6Q aQ aQ b ∆(a|ab)∆(a|bb)+i+ 3Q b Q b Q b ∆(a|aa)∆(aa|b) + 3Q b Q b Q b ∆(a|aa)∆(a|ab) − 3Q aQ b Q b ∆(a|aa)∆(ab|b) − 3Q aQ b Q b ∆(a|aa)∆(a|bb) ++∆(a|b)T aaaˆ− Qb Q b Q b ∆(a|b)˜ + T aabˆ3QaQ b Q b ∆(a|b)˜ + T abbˆ− 3QaQ aQ b ∆(a|b)˜ + T bbbˆQaQ aQ a∆(a|b)˜++∆(a|b)T aaˆ− 2Qb Q b Q ab ∆(a|b) − Q b Q b Q b ∆(aa|b) − Q b Q b Q b ∆(a|ab)++ 2Q aQ b Q b ∆(a|b) + Q aQ b Q b Q b ∆(ab|b) + Q aQ b Q b ∆(a|bb)˜++∆(a|b)T abˆ2Qb Q b Q aa∆(a|b) + 2Q aQ b Q ab ∆(a|b) + 2Q aQ b Q b ∆(aa|b) + 2Q aQ b Q b ∆(a|ab)−− 2Q aQ b Q ab ∆(a|b) − 2Q aQ aQ bb ∆(a|b) − 2Q aQ aQ b ∆(ab|b) − 2Q aQ aQ b ∆(a|bb)˜++∆(a|b)T bbˆ − 2QaQ b Q aa∆(a|b) − Q aQ aQ b ∆(aa|b) − Q aQ aQ b ∆(a|ab)++ 2Q aQ aQ ab ∆(a|b) + Q aQ aQ a∆(ab|b) + Q aQ aQ a∆(a|bb)˜++∆(a|b)T aaˆ− QaQ aQ b ∆(b|bb) + 2Q aQ b Q b ∆(b|ab) − Q b Q b Q b ∆(b|aa)˜++∆(a|b)T abˆQaQ aQ a∆(b|bb) − 2Q aQ aQ b ∆(b|ab) + Q aQ b Q b ∆(b|aa)˜++∆(a|b)T baˆQaQ aQ b ∆(a|bb) − 2Q aQ b Q b ∆(a|ab) + Q b Q b Q b ∆(a|aa)˜++∆(a|b)T bbˆ− QaQ aQ a∆(a|bb) + 2Q aQ aQ b ∆(a|ab) − Q aQ b Q b ∆(a|aa)˜++∆(a|b)T ah− 2Q aQ b Q aa∆(b|bb) − Q aQ aQ b ∆(ab|bb) − Q aQ aQ b ∆(b|abb)++ 2Q b Q b Q aa∆(b|ab) + 2Q aQ b Q ab ∆(b|ab) + 2Q aQ b Q b ∆(ab|ab) 0+ 2Q aQ b Q b ∆(b|aab)−− 2Q b Q b Q ab ∆(b|aa) − Q b Q b Q b ∆(ab|aa) − Q b Q b Q b ∆(b|aaa)++ 2Q aQ aQ ab ∆(b|bb) + Q aQ aQ a∆(bb|bb) 0+ Q aQ aQ a∆(b|bbb)−− 2Q aQ b Q ab ∆(b|ab) − 2Q aQ aQ bb ∆(b|ab) − 2Q aQ aQ b ∆(bb|ab) − 2Q aQ aQ b ∆(b|abb)+i+ 2Q aQ b Q bb ∆(b|aa) + Q aQ b Q b ∆(bb|aa) + Q aQ b Q b ∆(b|aab) ++∆(a|b)T bh2Q aQ b Q aa∆(a|bb) + Q aQ aQ b ∆(aa|bb) + Q aQ aQ b ∆(a|abb)−− 2Q b Q b Q aa∆(a|ab) − 2Q aQ b Q ab ∆(a|ab) − 2Q aQ b Q b ∆(aa|ab) 0− 2Q aQ b Q b ∆(a|aab)++ 2Q b Q b Q ab ∆(a|aa) + Q b Q b Q b ∆(aa|aa) + Q b Q b Q b ∆(a|aaa)−− 2Q aQ aQ ab ∆(a|bb) − Q aQ aQ a∆(ab|bb) 0− Q aQ aQ a∆(a|bbb)++ 2Q aQ b Q ab ∆(a|ab) + 2Q aQ aQ bb ∆(a|ab) + 2Q aQ aQ b ∆(ab|ab) + 2Q aQ aQ b ∆(a|abb)−i− 2Q aQ b Q bb ∆(a|aa) − Q aQ b Q b ∆(ab|aa) − Q aQ b Q b ∆(a|aab) .The simplification (collecting all terms) gives:G yxy xy x =j1 Taaaˆ− Q3b ∆(a|b) 2˜ + T aabˆ3QaQ 2 b∆(a|b) 2˜+[∆(a|b)] 5 + T abbˆ− 3Q2a Q b ∆(a|b) 2˜ + T bbbˆQ3 a ∆(a|b) 2˜+h+T aa − 2Q 2 bQ ab ∆(a|b) 2 + 2Q aQ b Q bb ∆(a|b) 2 + 3Q 3 b∆(a|b)∆(aa|b) + 2Q 3 b∆(a|b)∆(a|ab)−i− 4Q aQ 2 b∆(a|b)∆(ab|b) − 2Q aQ 2 b∆(a|b)∆(a|bb) − Q 2 aQ b ∆(a|b)∆(b|bb) +h+T ab − 2Q 2 aQ bb ∆(a|b) 2 + 2Q b Q b Q aa∆(a|b) 2 + Q 3 a∆(a|b)∆(b|bb) + 6Q 2 aQ b ∆(a|b)∆(ab|b)+i+ Q 3 b∆(a|b)∆(a|aa) − 6Q aQ 2 b∆(a|b)∆(a|ab) + 5Q 2 aQ b ∆(a|b)∆(a|bb) − 5Q aQ 2 b∆(a|b)∆(aa|b) +h+T bb − 2Q aQ b Q aa∆(a|b) 2 + 2Q 2 aQ ab ∆(a|b) 2 − 3Q 3 a∆(a|b)∆(a|bb) − 2Q 3 a∆(a|b)∆(ab|b)+i+ 4Q 2 aQ b ∆(a|b)∆(a|ab) + 2Q 2 aQ b ∆(a|b)∆(aa|b) − Q aQ 2 b∆(a|b)∆(a|aa) +

213+T ah3Q 2 a Q b∆(aa|b)∆(b|bb) + 3Q 2 a Q b∆(a|ab)∆(b|bb) − 3Q 3 a ∆(ab|b)∆(b|bb) − 3Q3 a ∆(a|bb)∆(b|bb)−− 6Q aQ 2 b ∆(aa|b)∆(b|ab) − 6QaQ2 b ∆(a|ab)∆(b|ab) + 6Q2 a Q b∆(ab|b)∆(b|ab) + 6Q 2 a Q b∆(a|bb)∆(b|ab)−− 3Q 3 b ∆(aa|b)∆(b|aa) − 3Q3 b ∆(a|ab)∆(b|aa) + 3QaQ2 b ∆(ab|b)∆(b|aa) + 3QaQ2 b ∆(a|bb)∆(b|aa)−− 2Q aQ b Q aa∆(a|b)∆(b|bb) + 2Q 2 b Qaa∆(a|b)∆(b|ab) + 2QaQ bQ ab ∆(a|b)∆(b|ab) − 2Q 2 b Q ab∆(a|b)∆(b|aa)−− Q 2 aQ b ∆(a|b)∆(ab|bb) − Q 2 aQ b ∆(a|b)∆(b|abb) + 2Q aQ 2 b ∆(a|b)∆(b|aab) − Q3 b ∆(a|b)∆(ab|aa) − Q3 b ∆(a|b)∆(b|aaa)++ 2Q 2 aQ ab ∆(a|b)∆(b|bb) − 2Q aQ b Q ab ∆(a|b)∆(b|ab) − 2Q 2 aQ bb ∆(a|b)∆(b|ab) + 2Q aQ b Q bb ∆(a|b)∆(b|aa)++ Q 3 a∆(a|b)∆(b|bbb) − 2Q 2 aQ b ∆(a|b)∆(bb|ab) − 2Q 2 aQ b ∆(a|b)∆(b|abb) + Q aQ 2 b ∆(a|b)∆(bb|aa) + QaQ2 bi+∆(a|b)∆(b|aab)+T bh3Q 2 a Q b∆(a|bb)∆(aa|b) + 3Q 2 a Q b∆(a|bb)∆(a|ab) − 3Q 3 a ∆(a|bb)∆(ab|b) − 3Q3 a ∆(a|bb)∆(a|bb)−− 6Q aQ 2 b ∆(a|ab)∆(aa|b) − 6QaQ2 b ∆(a|ab)∆(a|ab) + 6Q2 a Q b∆(a|ab)∆(ab|b) + 6Q aQ aQ b ∆(a|ab)∆(a|bb)++ 3Q 2 b ∆(a|aa)∆(aa|b) + 3Q2 b ∆(a|aa)∆(a|ab) − 3QaQ2 b ∆(a|aa)∆(ab|b) − 3QaQ2 b ∆(a|aa)∆(a|bb)++ 2Q aQ b Q aa∆(a|b)∆(a|bb) − 2Q 2 b Qaa∆(a|b)∆(a|ab) − 2QaQ bQ ab ∆(a|b)∆(a|ab) + 2Q 2 b Q ab∆(a|b)∆(a|aa)++ Q 2 aQ b ∆(a|b)∆(aa|bb) + Q 2 aQ b ∆(a|b)∆(a|abb) − 2Q aQ 2 b ∆(a|b)∆(a|aab) + Q3 b ∆(a|b)∆(aa|aa) + Q3 b ∆(a|b)∆(a|aaa)−− 2Q 2 aQ ab ∆(a|b)∆(a|bb) + 2Q aQ b Q ab ∆(a|b)∆(a|ab) + 2Q 2 aQ bb ∆(a|b)∆(a|ab) − 2Q aQ b Q bb ∆(a|b)∆(a|aa)−− Q 3 a∆(a|b)∆(a|bbb) + 2Q 2 aQ b ∆(a|b)∆(ab|ab) + 2Q 2 aQ b ∆(a|b)∆(a|abb) − Q aQ 2 b ∆(a|b)∆(ab|aa) − QaQ2 bi.∆(a|b)∆(a|aab)The full expansion of G yxy xy xy xwill not be presented here.REFERENCES[1] Bryant, R.L.: Élie Cartan and geometric duality, Journées Élie Cartan 1998 & 1999, Inst. É.Cartan 16 (2000), 5–20.[2] Cartan, É.: Sur les variétés à connexion projective, Bull. Soc. Math. France 52 (1924), 205–241.[3] Cartan, É.: Sur la géométrie pseudo-conforme des hypersurfaces de l’espace de deux variablescomplexes, II, Ann. Scuola Norm. Sup. Pisa 1 (1932), 333–354.[4] Christoffel, E.B.: Über die Transformation der homogenen Differentialausdrücke zweitenGrades, J. reine angew. Math. 70 (1869), 46–70.[5] Crampin, M.; Saunders, D.J.: Cartan’s concept of duality for second-order ordinary differentialequations, J. Geom. Phys. 54 (2005), 146–172.[6] Engel, F.; Lie, S.: Theorie der transformationsgruppen. Erster Abschnitt. Unter Mitwirkungvon Dr. Friedrich Engel, bearbeitet von Sophus Lie, B.G. Teubner, Leipzig, 1888. Reprintedby Chelsea Publishing Co. (New York, N.Y., 1970).[7] Grissom, C.; Thompson, G.; Wilkens, G.: Linearization of second-order ordinary differentialequations via Cartan’s equivalence method, J. Diff. Eq. 77 (1989), no. 1, 1–15.[8] Hsu, L.; Kamran, N.: Classification of second order ordinary differential equations admittingLie groups of fibre-preserving point symmetries, Proc. London Math. Soc. 58 (1989), no. 3,387–416.[9] Isaev, A.V.: Zero CR-curvature equations for rigid and tube hypersurfaces, Complex Variablesand Elliptic Equations, 54 (2009), no. 3-4, 317–344.[10] Koppisch, M.A.: Zur Invariantentheorie der gewöhlichen Differentialgleichungen zweiterOrdnung, Inaugural Dissertation, Leipzig, B.-G. Teubner, 1905.[11] Lie, S.: Klassifikation und Integration von gewöhnlichen Differentialgleichungen zwischenx, y, die eine Gruppe von Transformationen gestaten I-IV. In: Gesammelte Abhandlungen,Vol. 5, B.G. Teubner, Leipzig, 1924, pp. 240–310; 362–427, 432–448.[12] Lie, S.; Scheffers, G.: Vorlesungen über continuirlichen gruppen, mit geometrischen undanderen anwendungen, B.-G. Teubner, Leipzig, 1893. Reprinted by Chelsea Publishing Co.(New York, N.Y., 1971).[13] Merker, J.: Convergence of formal biholomorphisms between minimal holomorphically nondegeneratereal analytic hypersurfaces, Int. J. Math. Math. Sci. 26 (2001), no. 5, 281–302.[14] Merker, J.: On the partial algebraicity of holomorphic mappings between real algebraic sets,Bull. Soc. Math. France 129 (2001), no. 3, 547–591.

214[15] Merker, J.: On envelopes of holomorphy of domains covered by Levi-flat hats and the reflectionprinciple, Ann. Inst. Fourier (Grenoble) 52 (2002), no. 5, 1443–1523.[16] Merker, J.: On the local geometry of generic submanifolds of C n and the analytic reflectionprinciple, Journal of Mathematical Sciences (N. Y.) 125 (2005), no. 6, 751–824.[17] Merker, J.: Étude de la régularité analytique de l’application de réflexion CR formelle, AnnalesFac. Sci. Toulouse, XIV (2005), no. 2, 215–330.[18] Merker, J.: Explicit differential characterization of the Newtonian free particle system in m 2 dependent variables, Acta Mathematicæ Applicandæ, 92 (2006), no. 2, 125–207.[19] Merker, J.; Porten, E.: Holomorphic extension of CR functions, envelopes of holomorphy andremovable singularities, International Mathematics Research Surveys, Volume 2006, ArticleID 28295, 287 pages.[20] Merker, J.: Lie symmetries of partial differential equations and CR geometry, Journal of MathematicalSciences (N.Y.), to appear (2009), arxiv.org/abs/math/0703130[21] Merker, J.: Sophus Lie, Friedrich Engel et le problème de Riemann-Helmholtz, HermannÉditeurs, Paris, 2010, à paraître, 307 pp., arxiv.org/abs/0910.0801[22] Merker, J.: Vanishing Hachtroudi curvature and spherical real analytic hypersurfaces,arxiv.org, to appear.[23] Nurowski, P.; Sparling, G.A.J.: 3-dimensional Cauchy-Riemann structures and 2 nd orderordinary differential equations, Class. Quant. Gravity, 20 (2003), 4995–5016.[24] Segre, B.: Intorno al problema di Poincaré della rappresentazione pseudoconforme, Rend.Acc. Lincei, VI, Ser. 13 (1931), 676–683.[25] Tresse, A.: Détermination des invariants ponctuels de l’équation différentielle du secondordre y ′′ = ω(x,y,y ′ ), Hirzel, Leipzig, 1896.

215Vanishing Hachtroudi curvatureand local equivalenceto the Heisenberg pseudosphereJoël MerkerAbstract. To any completely integrable second-order system of real or complexpartial differential equations:(y x k 1x k 2 = F k1 ,k 2 x 1 ,... ,x n ), y, y x 1,... ,y x nwith 1 k 1 , k 2 n and with F k1 ,k 2= F k2 ,k 1in n 2 independent variables(x 1 ,...,x n ) and in one dependent variable y, Mohsen Hachtroudi associated in1937 a normal projective (Cartan) connection, and he computed its curvature. Bymeans of a natural transfer of jet polynomials to the associated submanifold ofsolutions, what the vanishing of the Hachtroudi curvature gives can be preciselytranslated in order to characterize when both families of Segre varieties and of conjugateSegre varieties associated to a Levi nondegenerate real analytic hypersurfaceM in C n (n 3) can be straightened to be affine complex (conjugate) lines. Incontinuation to a previous paper devoted to the quite distinct C 2 -case, this thencharacterizes in an effective way those hypersurfaces of C n+1 in higher complexdimension n + 1 3 that are locally biholomorphic to a piece of the (2n + 1)-dimensional Heisenberg quadric, without any special assumption on their definingequations.arxiv.org/abs/0910.2861/Table of contents1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.2. Segre varieties and differential equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .220.3. Geometry of associated submanifolds of solutions. . . . . . . . . . . . . . . . . . . . . . . . .224.4. Effective differential characterization of pseudosphericality in C n+1 . . . . . . 226.§1. INTRODUCTIONThe explicit characterization of pseudosphericality of an arbitrary real analyticlocal hypersurface sitting in the complex Euclidean space has been(re)studied recently by Isaev in [11], who employed the famous Chern(-Moser) tensorial approach [CM1974, Ch1975] to the concerned equivalenceproblem. But in the growing literature devoted to Lie-group symmetriesof Cauchy-Riemann manifolds, only a very few articles underline that, alreadyin his 1937 Ph.D. thesis [Ha1937] under the direction of his Élie Cartan— who was around the same period also the master of Chern —, the

216Iranian mathematician Mohsen Hachtroudi (cited briefly only in [Ch1975])constructed directly an explicit normal projective Cartan connection canonicallyassociated to any completely integrable system of real or complexpartial differential equations:( )(7.28) y x k 1x k 2 (x) = F k1 ,k 2 x 1 , . . .,x n , y, y x 1, . . ., y x nin n 2 independent variables x 1 , . . .,x n and in one dependent variabley, by endeavouring in a successful way to generalize the celebratedpaper [Ca1924]. Chern’s clever observation in 1974 that Hachtroudi’s37 years-old approach was intrinsically related to the nascent higherdimensionalCR geometry was followed, in his two papers in question, byhis technical contribution of redoing (only) parts of Hachtroudi’s effectivecomputations, following the alternative (heavier, though essentially equivalent)strategy of constructing a posteriori the projective connection, afterhaving reinterpreted at the beginning the problem in terms of the wide andpowerful Cartan Method of Equivalence. Thus, one should be aware, historicallyspeaking, that in the original reference [Ha1937], much more completegeometric and computational aspects were published long before, thoughthey were expressed in a purely analytic and somewhat elliptic languagewhich, unfortunately for us at present times, does not transmit in words andwith figures all the underlying geometric meanings which were clear then toÉlie Cartan.Because Hachtroudi was able to write down explicitly his curvature tensors,he deduced the second-order system (7.28) — below — of partialdifferential equations that the functions F k1 ,k 2should satisfy in order thatthe system (7.28) be equivalent, through a point transformation (x, y) ↦→(x ′ , y ′ ) = ( x ′ (x, y), y ′ (x, y) ) to the simplest system: y ′ x ′k 1x ′k 2 (x′ ) = 0, withall right-hand sides being zero. In the present article, a companion and afollower of a preceding one [22] devoted to the quite different C 2 -case, wewill apply, to the higher-dimensional characterization of pseudosphericality,this effective necessary and sufficient condition (7.28) due to Hachtroudiwhich, however and inexplicably, is totally inextant in the two contributionsof Chern. We hope in this way to complete the explicit characterization ofpseudosphericality for rigid or even tube hypersurfaces that was obtained recentlyby Isaev in [11], because apparently, the general (nonrigid) case wasstill open in the specialized field.We now start the exposition. Let M be a local real analytic in C n+1 .Though the basic definitions, lemmas and propositions of the theory arevalid in any complex dimension n + 1 2, there is a strong computationaldifference between the two characterizations of sphericality for n = 1 (compare[21]) and of pseudosphericality for n 2 (presently), so that, in order

to fix the ideas, it will be assumed throughout the paper — and recalledwhen necessary — that the CR dimension n is always 2.Locally in a neighborhood of one of its points p, the hypersurface M maybe represented, in any system of local holomorphic coordinates:t = (w, z) ∈ C n × Cvanishing at p for which the w-axis is not complex-tangent to M at p, by aso-called complex defining equation — Section 2 provides further informations— of the form:(7.28) w = Θ ( z, z, w) = Θ ( z, t ) ,or equivalently in a more expanded form which exhibits all the indices:w = Θ ( z 1 , . . .,z n , z 1 , . . ., z n , w ) = Θ ( z 1 , . . ., z n , t 1 , . . .,t n , t n+1).Then M localized at p is called pseudospherical (at p) if it is biholomorphicto a piece of one Heisenberg pseudosphere:(7.28) Im w ′ = |z ′ 1 |2 + · · · + |z ′ q |2 − |z ′ q+1 |2 − · · · − |z ′ n |2 ,for some q with 0 q n, the number of positive eigenvalues of thenondegenerate Levi form. Next, let us introduce the following Jacobian-likedeterminant:∆ :=∣Θ z1 · · · Θ zn Θ wΘ z1 z 1· · · Θ z1 z nΘ z1 w·· · · · ·· ··Θ znz 1· · · Θ znz nΘ znw=∣ ∣217∣Θ t1· · · Θ tnΘ tn+1∣∣∣∣∣∣∣Θ z1 t 1· · · Θ z1 t nΘ z1 t n+1.·· · · · ·· ··Θ znt 1· · · Θ znt nΘ zntn+1 For any index µ ∈ {1, . . ., n, n + 1} and for any index l ∈ {1, . . ., n}, letalso ∆ µ [0 1+l ]denote the same determinant, but with its µ-th column replacedby the transpose of the line (0 · · ·1 · · ·0) with 1 at the (1+l)-th place, and 0elsewhere, its other columns being untouched. One easily convinces oneself(but see also Section 2) that M is Levi-nondegenerate at p — which is theorigin of our system of coordinates — if and only if ∆ does not vanish at theorigin, whence ∆ is nowhere zero in some sufficiently small neighborhoodof the origin. Similarly, for any indices µ, ν, τ ∈ {1, . . .,n, n + 1}, denoteby ∆ τ the same determinant as ∆, but with only its τ-th column replaced[t µ t ν ]by the transpose of the line:( )Θt µ ν t Θ z1 t µ t ν · · · Θ znt µ t ν ,other columns being again untouched. All these determinants ∆, ∆ µ [0 1+l ] ,∆ τ [t µ t ν ]are visibly universal differential expressions depending upon thesecond-order jet J 2 z,z,w Θ and upon the third-order jet J3 z,z,w Θ.Main Theorem. An arbitrary, not necessarily rigid, real analytic hypersurfaceM ⊂ C n+1 with n 2 which is Levi nondegenerate at one of its points

218p and has a complex defining equation of the form (7.28) in some systemof local holomorphic coordinates t = (z, w) ∈ C n × C vanishing at p, ispseudospherical at p if and only if its complex graphing function Θ satisfiesthe following explicit nonlinear fourth-order system of partial differentialequations:0 ≡n+1Xn+1Xµ=1 ν=1− δ k 1 ,l 1n+2− δ k 1 ,l 2n+2− δ k 2 ,l 1n+2− δ k 2 ,l 2n+2»j∆ µ [0 1+l1 ] · ∆ν [0 1+l2 ] ∆ ·nX∆ µ [0 1+l ′] · ∆ν [0 1+l2 ]l ′ =1nX∆ µ [0 1+l1 ] · ∆ν [0 1+l ′]l ′ =1nX∆ µ [0 1+l ′] · ∆ν [0 1+l2 ]l ′ =1nX∆ µ [0 1+l1 ] · ∆ν [0 1+l ′]l ′ =11j∆ ·j∆ ·j∆ ·j∆ ·+· ˆδ (n+1)(n+2) k1 ,l 1δ k2 ,l 2+ δ k2 ,l 1δ k1 ,l 2˜·nX nXj· ∆ µ [0 1+l ′] · ∆ν [0 1+l ′′] ∆ ·l ′ =1 l ′′ =1∂ 4 n+1Θ X− ∆ τ [t∂z k1 ∂z k2 ∂t µ∂t µ t ν ] ·ντ=1∂ 4 n+1Θ X− ∆ τ [t∂z l ′∂z k2 ∂t µ∂t µ t ν ] ·ντ=1∂ 4 n+1Θ X− ∆ τ [t∂z l ′∂z k2 ∂t µ∂t µ t ν ] ·ντ=1∂ 4 n+1Θ X− ∆ τ [t∂z k1 ∂z l ′∂t µ∂t µ t ν ] ·ντ=1∂ 4 n+1Θ X− ∆ τ [t∂z k1 ∂z l ′∂t µ∂t µ t ν ] ·ντ=1∂ 4 n+1Θ X− ∆ τ [t∂z l ′∂z l ′′∂t µ∂t µ t ν ] ·ντ=1ff∂ 3 Θ∂z k1 ∂z k2 ∂t τ −ff∂ 3 Θ∂z l ′∂z k2 ∂t τ −ff∂ 3 Θ∂z l ′∂z k2 ∂t τ −ff∂ 3 Θ∂z k1 ∂z l ′∂t τ −ff∂ 3 Θ∂z k1 ∂z l ′∂t τ +ff∂ 3 Θ∂z l ′∂z l ′′∂t τ ,for all pairs of indices (k 1 , k 2 ) with 1 k 1 , k 2 n, and for all pairs ofindices (l 1 , l 2 ) with 1 l 1 , l 2 n.The written system is effective: no implicit formal expression is involvedand pseudosphericality is characterized directly and only in terms of Θ.Now, here is a summarized description of our arguments of proof. Abit similarly as for the C 2 -case — but with major differences afterwards —which was already studied in [21], we may associate to any such Levinondegenerate real analytic local hypersurface M ⊂ C n+1 of equationw = Θ(z, z, w) a uniquely defined system of second-order partial differentialequations:(7.28) w zk1 z k2(z) = Φ k1 ,k 2(z, w(z), wz (w) ) (1 k 1 , k 2 n)with Φ k1 ,k 2= Φ k2 ,k 1, simply by eliminating the two variables z and w,viewed as parameters, from the set of n + 1 equations 23 :w(z) = Θ ( z, z, w ) ( ) ( ), w z1 (z) = ∂Θ∂z 1z, z, w , . . ...., wzn (z) = ∂Θ∂z nz, z, w ,— the assumption that the Jacobian determinant ∆ is nonvanishing atthe origin being precisely the one which guarantees, technically speaking,that the classical (holomorphic) implicit function theorem applies — and23 This process appears for instance in the references [8, Ha1937, Ch1975, Su2001,Su2002, 1, 19].

then by replacing the(so obtained)values for z and w in all second orderderivatives2 Θ ∂∂z k1 z k2z, z, w , see (7.28) below. Trivially, this systemis completely integrable, for we just derived it from its general solutionw(z) := Θ ( z, z, w ) , where (z, w) are understood as parameters.As we said, Hachtroudi showed that the curvature of the projective normal(Cartan) connection he associated with the system (7.28) vanishes if andonly if the right-hand side functions F k1 ,k 2satisfy the following explicit differentialsystem, which is linear in terms of their second-order derivatives(all of which, notably, appear only with respect to the y x l):(7.28)0 ≡ ∂2 F k1,k 2∂y x l 1 y x l 2−− 1n+2n∑l ′ =1(δ k1,l 1∂ 2 F l′ ,k 2∂y x l ′∂y x l 2[ ] ∑n 1+(n+1)(n+2) δk1,l 1δ k2,l 2+ δ k2,l 1δ k1,l 2l ′ =1∂ 2 F l′ ,k+ δ2∂ 2 F k1,lk1,l 2+ δ ′k2,l∂y x l 1 ∂y1x l ′ ∂y ′∂y x l x l 2n∑l ′′ =1219∂ 2 )F k1,l+ δ ′k2,l 2+∂y x l 1 ∂y x l ′∂ 2 F l ′ ,l ′′∂y x l ′∂y x l′′ (1 k 1, k 2 n)(1 l 1, l 2 n) .Hachtroudi also showed that this latter condition, better known nowadaysamongst the Several Complex Variables community as vanishing ofChern(-Moser) curvature to which it indeed amounts, characterizes thelocal equivalence, through a point transformation (x, y) ↦→ (x ′ , y ′ ) =(x ′ (x, y), y ′ (x, y) ) , to the simplest system: y ′ x ′k 1x ′k 2 (x′ ) = 0. We thenremind the semi-known fact that M is pseudospherical if and only if itsassociated second-order system (7.28) is equivalent, through a local biholomorphism(z, w) ↦→ (z ′ , w ′ ) = ( z ′ (z, w), w ′ (z, w) ) fixing the origin, to thesimplest system w ′ z ′ k 1z ′ k 2(z ′ ) = 0. So we may apply to the functions Φ k1 ,k 2Hachtroudi’s vanishing curvature equations (7.28), but still, the Φ k1 ,k 2arenot expressed in terms of Θ, for they were constructed by employing someunpleasant implicit functions when solving above for z and w. Fortunately,here similarly as in [21], we may apply the techniques of computational differentialalgebra sketched in [19] in order to explicitly express any algebraicexpressions in the second-order jet of the Φ k1 ,k 2in terms of the fourth-orderjet of Θ, and the appropriate general equation which we shall need:∂ 2 Φ k1,k 2= 1 n+1∑∂w zl1 ∂w zl2 ∆ 3n+1∑µ=1 ν=1∆ µ [0 1+l1 ]· ∆ν [0 1+l2 ]{∆ ·∂ 4 n+1Θ ∑∂z k1 ∂z k2 ∂t µ ∂t ν −τ=1∆ τ [t µ t ν ] ·will be obtained in Section 4 below, after rather lengthy but elementary calculations,parts of which are inspired from [17]. It is now essentially clearhow one obtains the (boxed) long fourth-order differential equations statedin the theorem, but in any case, some complete details will be provided atthe very end of the paper.}∂ 3 Θ∂z k1 ∂z k2 ∂t τ

220To conclude this extensive introduction which was designed for readerswanting to quickly embrace the contents, we would like to draw theattention on the work [22], whose manual calculations where finalized inmanuscript form already in 2003 24 , and which will soon confirm the abovetheorem by following another route, viz. by calculating explicitly the socalledChern(-Moser) tensor differential forms, which might interest somecontemporary CR geometers better than the (essentially equivalent) originalCartan-Hachtroudi(-Tanaka) approach.§2. SEGRE VARIETIES AND DIFFERENTIAL EQUATIONSReal analytic hypersurfaces in C n+1 . Let us therefore consider an arbitraryreal analytic hypersurface M in C n+1 with n 2, and let us localizeit around one of its points, say p ∈ M. Then there exist complex affinecoordinates:(z, w) = (z 1 , . . .,z n , w) = ( x 1 + √ −1y 1 , . . .,x n + √ −1 y n , u+iv ) = ( x+ √ −1y, u+ √ −1 v )vanishing at p in which T p M = {u = 0}, so that M is represented in aneighborhood of p by a graphed defining equation of the form:u = ϕ(x, y, v) = ϕ ( x 1 , . . .,x n , y 1 , . . .,y n , v ) ,where the real-valued function:∑ϕ = ϕ(x, y, v) =k∈N n , l∈N n , m∈N|k|+|l|+m2ϕ k,l,m x k y l v m ∈ R { x, y, u } ,which possesses entirely arbitrary real coefficients ϕ k,l,m , vanishes at theorigin: ϕ(0) = 0, together with all its first order derivatives: 0 = ∂ x kϕ(0) =∂ y lϕ(0) = ∂ v ϕ(0). By simply rewriting this initial real equation of M as:w+w= ϕ ( z+z, z−z2 2 2 √ , w−w−1 2 −1) √ ,and then by solving the so written equation with respect to w, one obtainsan equation of the shape:w = Θ ( z, z, w ) ∑= Θ k,l,m z k z l w m ∈ C { z, z, w } ,k∈N n , l∈N n , m ∈ N|k|+|l|+m1whose right-hand side converges of course near the origin (0, 0, 0) ∈ C n ×C n × C and whose coefficients Θ k,l,m ∈ C are complex. Since dϕ(0) = 0,one has Θ = −w + order 2 terms.The paradox that any such complex equation provides in fact two realdefining equations for the real hypersurface M which is one-codimensional,24 At the conference Cauchy-Riemann Analysis and Geometry organized by Ingo Lieband Gerd Schmalz at the Max-Planck Institut of Bonn, 22–27 September 2003, the authorgave a talk the title of which was “Explicit Chern-Moser tensors”.

and also in addition the fact that one could as well have chosen to solve theabove equation with respect to w, instead of w, these two apparent “contradictions”are corrected by means of a fundamental, elementary statementthat transfers to Θ (in a natural way) the condition of reality:∑∑ϕ(x, y, u) = ϕ k,l,m x k y l v m = ϕ k,l,m x k y l v m = ϕ(x, y, v)|k|+|l|+m1|k|+|l|+m1enjoyed by the initial definining function ϕ. In the sequel, we shall workexclusively with Θ; the reader is referred to [21] for justifications and motivations.Theorem. ([18], p. 19) The complex analytic function Θ = Θ(z, z, w) withΘ = −w + O(2) together with its complex conjugate:Θ = Θ ( ∑z, z, w) = Θ k,l,m z k z l w m ∈ C { z, z, w }k∈N n , l∈N n , m∈Nsatisfy the two (equivalent by conjugation) functional equations:(7.28)w ≡ Θ ( z, z, Θ(z, z, w) ) ,w ≡ Θ ( z, z, Θ(z, z, w) ) .Conversely, given a local holomorphic function Θ(z, z, w) ∈ C{z, z, w},Θ = −w + O(2) which, in conjunction with its conjugate Θ(z, z, w), satisfiesthis pair of equivalent identities, then the two zero-sets:{ ( )} { ( )}0 = −w + Θ z, z, w and 0 = −w + Θ z, z, wcoincide and define a local one-codimensional real analytic hypersurfaceM passing through the origin in C n+1 .Levi nondegeneracy. Within the hierarchy of nondegeneracy conditionsfor real hypersurfaces initiated by Diederich and Webster ([DW1980], seealso [Me2005a, Me2005b] for generalizations and a unification), Levi nondegeneracyis the most studied. The classical definition may be foundin [Bo1991] and in the survey of Chirka [Ch1991], but the following basicequivalent characterization can also be understood as a definition in thepresent paper. One may show ([Me2005a, Me2005b, 18]) that it is biholomorphicallyinvariant.Lemma. ([18], p. 28) The real analytic hypersurface M ⊂ C n+1 with0 ∈ M represented in coordinates (z 1 , . . .,z n , w) by a complex definingequation of the form w = Θ(z, z, w) is Levi nondegenerate at the origin ifand only if the map:(z1 , . . ., z n , w ) ↦−→(Θ ( 0, z, w ) ,∂Θ∂z 1(0, z, w), . . .,∂Θ∂z n(0, z, w) )221

222has nonvanishing (n+1) ×(n+1) Jacobian determinant at (z, w) = (0, 0).It follows then that this Jacobian determinant, not restricted to the origin:(7.28) ∆ = ∆ ( z, z, w ) Θ z1 · · · Θ zn Θ w:=Θ z1 z 1· · · Θ z1 z nΘ z1 w·· · · · ·· ··∣ Θ znz 1· · · Θ znz nΘ znw∣does not vanish in some small neighborhood of the origin in C n × C n × C.Levi nondegeneracy at the central point, i.e. ∆ ≠ 0 locally, will be assumedthroughout the present paper.Associated system of partial differential equations. At least since thepublication in 1888 by Lie and Engel in Leipzig of the Theorie der Transformationsguppen,it is known in a very general context — see Chapter 10of [8] and also [23, Ha1937, Ch1975, Fa1980, 19, 1, 21] — that, to thewhole family of Segre varieties:S z,w := { (z, w) ∈ C n × C: w = Θ ( z, z, w )}parametrized by the n + 1 antiholomorphic variables ( z 1 , . . ., z n , w ) , onemay canonically associate a completely integrable second-order system ofpartial differential equations whose general solution is precisely the functionΘ ( z, z, w ) . Indeed, considering w as a function w = w(z) of (z 1 , . . .,z n )in the defining equation of M, one differentiates it once with respect to eachvariable z 1 , . . .,z n so that one gets the n + 1 equations:w(z) = Θ ( z, z, w ) ,w z1 (z) = ∂Θ∂z 1(z, z, w), . . ...., wzn (z) = ∂Θ∂z n(z, z, w).Then by means of the implicit function theorem — which applies preciselythanks to the nonvanishing of ∆ —, one may clearly solve for the n + 1antiholomorphic “parameters” (z, w), and this procedure provides a representation:z 1 = ζ 1(z, w(z), wz (z) ) , . . ., z n = ζ n(z, w(z), wz (z) ) , w = ξ ( z, w(z), w z (z) )with certain n + 1 uniquely defined local complex analytic functionsζ i (z, w, w z ) and ξ(z, w, w z ) of 2n + 1 complex variables. Utilizing thesefunctions, one is then pushed to replace z and w in all possible second-orderderivative:( )w zk1 z k2(z) =∂2 Θ∂z k1 ∂z k2z, z, w( ((7.28) = ∂2 Θ∂z k1 ∂z k2z, ζ z, w(z), wz (z) ) , ξ ( z, w(z), w z (z) ))=: Φ k1 ,k 2(z, w(z), wz (z) ) (k 1 , k 2 =1··· n),

and this defines without ambiguity the associated system of partial differentialequations. It is of second order. It is complete: all second-orderderivatives are functions of derivatives of lower order 1. In a sense to beprecised right now, it is also completely integrable because by construction,its general solution is Θ ( z, z, w ) .Geometric characterization of pseudosphericality. It is well known thatthe unit sphere:S 2n+1 = { (z 1 , . . ., z n , w) ∈ C n × C: |z 1 | 2 + · · · + |z n | 2 + |w| 2 = 1 }in C n minus one of its points, for instance: S 2n+1 \ {p ∞ } with p ∞ :=(0, . . ., 0, −1), is biholomorphic, through the so-called Cayley transform:(z 1 , . . .,z n , w) ↦−→ ( i z 11+w , . . ., i z n1+w , 1−w2+2 w)=: (z′1 , . . ., z ′ n , w′ )having inverse:(z ′ 1, . . ., z ′ n, w ′ ) ↦−→ ( −2iz ′ 11+2w ′ , . . ., −2iz′ n1+2w ′ ,1−2w ′1+2 w ′ )= (z1 , . . ., z n , w)to the so-called standard Heisenberg sphere of equation:w ′ = −w ′ + z ′ 1 z′ 1 + · · · + z′ n z′ nwhich sits in the target space (z ′ , w ′ ). Hence in the particular case when theLevi form of M has only positive eigenvalues, namely when q = n in (7.28),it follows clearly that M is spherical in the sense given in the Introduction ifand only if there exists a nonempty open neighborhood U 0 of 0 in C n+1 suchthat M ∩ U 0 is biholomorphic to a piece of the unit sphere. In general, thereare n − q negative eigenvalues in the Levi form, and this justifies adding a“pseudo”.Proposition. A Levi nondegenerate local real analytic hypersurface M inC n+1 is locally biholomorphic to a piece of the Heisenberg pseudosphere(hence pseudospherical) if and only if its associated second-order ordinarycomplex differential equation is locally equivalent to the second-order system:w z ′ k ′ z (z ′ ) = 0′ (1 k 1 , k 2 n),1 k 2with identically vanishing right-hand side.Proof. The n = 1 case, treated in great details by a previous reference [21],generalizes here with rather evident adaptations, hence will be skipped. Asn 2 throughout the present paper, one may also argue by slicing C n+1 byall possible copies of C 2 which pass through the origin and which containthe w-axis, so as to be able to apply the alreaday detailed n = 1 case.Geometrically, the local equivalence of M to the Heisenberg pseudospheremeans that, through some suitable local biholomorphism (z, w) ↦→223

224(z ′ , w ′ ) fixing the origin, both its Segre varieties and its conjugate Segrevarieties ([Me2005a, Me2005b, 18]):S z,w := { (z, w): w = Θ ( z, z, w )} and S z,w := { (z, w): w = Θ ( z, z, w )}are mapped to the Segre and conjugate Segre varieties of the Heisenbergpseudosphere:S ′ z ′ ,w = { w ′ = −w ′ + z ′ z ′} {and w ′ = −w ′ + z ′ z ′}′which, visibly, are plain complex affine lines.§3. GEOMETRY OF ASSOCIATED SUBMANIFOLDS OF SOLUTIONSCompletely integrable systems of partial differential equations. Thecharacterization of pseudosphericality we are dealing with holds in a contextmore general than just CR geometry 25 . Accordingly, let K denote eitherthe field C of complex numbers or the field R of real numbers, letx = (x 1 , . . ., x n ) ∈ K n with again n 2 — since the case n = 1 was alreadystudied in [21] —, let y ∈ K, and consider a system of the form (7.28).We will assume that it is completely integrable in the sense that the naturalcommutativity of partial derivatives enjoyed trivially by the left-hand sides:∂ 2 y x k 1x k 2/∂yx k 3 = ∂ 2 y x k 1x k 3/∂yx k 2(1 k 1 , k 2 , k 3 n)imposes immediately to the right-hand side functions F k1 ,k 2that they satisfythe so-called compatibility conditions:D k3(Fk1 ,k 2)= Dk2(Fk1 ,k 3),where we have introduced the following n total differentiation operators:D k := ∂∂x k + y x kn∑∂ + ∂y(1 k n)l=1F k,l∂∂y x lliving on the first-order jet space (x 1 , . . .,x n , y, y x 1, . . .,y x n). One verifiesthat these compatibility conditions amount to the fact that the n-dimensionaltangential distribution spanned by D 1 , . . .,D n in the (2n + 1)-dimensionalfirst-order [ ] jet space satisfies the classical Frobenius integrability conditionDk ′, D k ′′ = 0, and then the Clebsch-Frobenius theorem tells us thatthis distribution comes from a local foliation by n-dimensional manifoldsgraphed over the x-space that are naturally parametrized by n + 1 auxiliaryconstants (transversal directions) — call them a 1 , . . ., a n , b ∈ K —, namely25 We will be very brief here, the reader being referred to [19, 21] for the general theoreticalconsiderations.

the leaves of this local foliation may be explicitly represented as sets of theshape:{ (x 1 , . . .,x n , Q ( x 1 , . . .,x n , a 1 , . . ., a n , b ) ,225S 1( x 1 , . . ., x n , a 1 , . . ., a n , b ) , . . .,S n( x 1 , . . .,x n , a 1 , . . .,a n , b ))} ,where x 1 , . . .,x n vary freely and where Q, S 1 , . . ., S n are certain graphingfunctions. In fact, the functions S k are the first-order derivatives:S 1 = Q x 1, . . ...., S n = Q x nof the function Q, because by definition the integral curves of every vectorfield D k must be contained in such leaves, so that one has:∂Q= y∂x k x∣k any leaf= S kand furthermore also:∂S l= F k,l∣ ∣∂x k any leaf,whence we see that the fundamental graphing function Q = Q(x, a, b) happensto be the general solution to the initially given system of partial differentialequations:(Q x k 1x k 2 (x, a, b) ≡ F k1 ,k 2 x, Q(x, a, b), Qx 1(x, a, b), . . .,Q x n(x, a, b) )(1 k 1 , k 2 n).In the CR case, the fundamental function which is the general solution tothe associated system of partial diffential equations (7.28) is obviously thecomplex defining function Θ ( z, z, w ) , where the n + 1 quantities (z, w),viewed as independent variables, play the role of the constants (a, b).As in the n = 1 case, the constants (a 1 , . . .,a n , b) are best interpreted asa set of n + 1 initial conditions ( y x 1(0), . . .,y x n(0), −y(0) ) or integrationconstants, so that we can assume without loss of generality that the firstorderterms in the fundamental function Q are 26 :Q(x, a, b) = −b + x 1 a 1 + · · · + x n a n + O(|x| 2 ).It is then clear that the map:(a 1 , . . .,a n , b ) ↦−→ ( Q(0, a, b), Q x 1(0, a, b), . . .,Q x n(0, a, b) )(7.28)= ( − b, a 1 , . . .,a n)is of rank n + 1 at the origin, and this property remains also true whateverone chooses as a fundamental function Q(x, a, b), that is to say, withoutnecessarily assuming it to be normalized as above, which amounts to saying26 We put a minus sign in front of y(0) so as to match up with our choice of complexdefining equation w = − w + O(2).

226that 27 , in the parameter (a, b)-space, everything holds invariantly up to anylocal K-analytic transformation (a, b) ↦→ (a ′ , b ′ ) which does not involve thevariables (x, y).The way how one recovers the system of partial differential equations isvery similar to what we did in the CR case (7.28). Suppose indeed a bit moregenerally that we are given any local K-analytic function Q = Q(x, a, b)having the property that its first-order x-jet map (7.28) is of rank n + 1 at(a, b) = (0, 0). Then in the n + 1 equations:y(x) = Q(x, a, b), y x 1 = Q x 1(x, a, b), . . ...., Q x n(x, a, b),we can solve, by means of the implicit function theorem, for the n + 1constants (a 1 , . . .,a n , b), and this yields a representation:a k = A k( )x 1 , . . .,x n , y, y x 1, . . .,y x nb = B ( )x 1 , . . .,x n , y, y x 1, . . .,y x nfor certain functions A 1 , . . .,A n , B of (2n+1) variables. Then by replacingthese obtained values for the a k and for b in all the possible second-orderderivatives:y x k 1x k 2 = Q x k 1x k 2 (x, a, b)= Q x k 1x k 2(x, A(x, y, yx ), B(x, y, y x ) )= F k1 ,k 2(x, y, yx)it is rigorously clear that one may only recover the functions F k1 ,k 2westarted with.§4. EFFECTIVE DIFFERENTIAL CHARACTERIZATIONOF PSEUDOSPHERICALITY IN C n+1The 2n + 1 coordinates of the transformation considered at the moment:( ((7.28) x 1 , . . .,x n , y, y x 1, . . .,y x n)↦−→ x 1 , . . .,x n , a 1 , . . .,a n , b )and those of its inverse are given by the collection of functions:⎡⎡x j = x jx j = x j⎢⎣ a k = A k( x 1 ,...,x n ) ⎢,y,y x 1,... ,y x nb = B ( and ⎣ y = Q ( x 1 ,...,x n ,a 1 ,...,a n ,b )x 1 ,... ,x n ),y,y x 1,... ,y x n y x k = Q x k(x 1 ,... ,x n ,a 1 ,... ,a n ,b ) .For uniformity and harmony, we shall admit by convention the equivalencesof notation:b ≡ a n+1 and B ≡ A n+1 .27 Much more theoretical information is provided in [19].

Then by differentiating with respect to y x l each one of the following n + 1identically satisfied equations:y ≡ Q ( x 1 ,... ,x n , A 1( x 1 ,... ,x n ,y,y x 1,... ,y x n),...,A n( x 1 ,... ,x n ,y,y x 1,...,y x n),An+1 ( x 1 ,... ,x n )),y,y x 1,...,y x ny x k ≡ Q x k(x 1 ,... ,x n , A 1( x 1 ,... ,x n ,y,y x 1,...,y x n),... ,227A n( x 1 ,... ,x n ,y,y x 1,... ,y x n),An+1 ( x 1 ,... ,x n ,y,y x 1,... ,y x n)),we get the following n + n 2 equations:0 ≡ Q a 1 ∂A1 + · · · + Q ∂y a n ∂An + Q ∂An+1x l ∂y an+1 x l ∂y x lδ k,l = Q x k a 1 ∂A1 + · · · + Q ∂y x x l k a n ∂An + Q ∂y x x l k ∂An+1an+1 ∂y x l(k, l =1··· n).Fixing any l ∈ {1, . . ., n}, thanks to the assumption (Levi nondegeneracy)that the Jacobian determinant:∣ Q□ = □ ( a 1 · · · Q a n Q a n+1 ∣∣∣∣∣∣∣a 1 | · · · |a n |a n+1) Q:=x 1 a 1 · · · Q x 1 a n Q x 1 a n+1. . ..,. .∣Q x n a 1 · · · Q x n a n Q x n a n+1does not vanish, we may solve — just by means of Cramer’s rule — for then + 1 unknowns ∂Aµ , the above system of n + 1 equations, and this gives∂y x lus:(7.28)∂A µ∂y x l= □ [0 µ 1+l ]□ := □(a 1| · · · |a µ−1 |0 1+l |a µ+1 | · · · |a n+1 )□(a 1 | · · · |a µ−1 |a µ |a µ+1 | · · · |a n+1 ) ,where 0 1+l is a specific notation to denote the column consisting of n + 1zeros piled up, except at the (1 + l)-th level from its top, where instead of 0,one reads 1, and where, as our notation with vertical bars helps to guess::=∣□ µ [0 1+l ] = □(a 1| · · · |a µ−1 | µ 0 1+l |a µ+1 | · · · |a n+1 ) :=∣Q a 1 · · · Q a µ−1 0 Q a µ+1 · · · Q a n+1 ∣∣∣∣∣∣∣∣∣∣∣Q x 1 a 1 · · · Q x 1 a µ−1 0 Q x 1 a µ+1 · · · Q x 1 a n+1·· · · · ·· ·· ·· · · · ··Q x k a 1 · · · Q x k a µ−1 1 Q x k a µ+1 · · · Q .x k a n+1·· · · · ·· ·· ·· · · · ··Q x n a 1 · · · Q x n a µ−1 0 Q x n a µ+1 · · · Q x n a n+1To avoid any ambiguity, we shall sometimes put the integer µ in the upperindex position of the vertical bar to indicate precisely which column is concerned.As is clear, this notation allows one to view and to remember what

228are the involved partial derivatives of the fundamental function Q that appearinside each column. In summary, □ µ [0 1+l ]comes from □ by changingjust its µ-th column, as Cramer’s rule classically says.Next, the two-ways transfer between local functions G defined in the(x, y, y x )-space and local functions T defined in the (x, a, b)-space, namelythe one-to-one correspondence:G ( ) (x 1 , . . ., x n , y, y x 1, . . .,y x n ←→ T x 1 , . . .,x n , a 1 , . . ., a n , b )through the diffeomorphism (7.28), may be viewed concretely, in the directionwe are interested in, as the following identity:G ( x 1 , . . .,x n , y, y x 1, . . .,y x n)≡≡ T ( x 1 , . . .,x n ,A 1( x 1 , . . ., x n , y, y x 1, . . .,y x n) ( , . . .,Anx 1 , . . .,x n , y, y x 1, . . .,y x n),A n+1( ))x 1 , . . .,x n , y, y x 1, . . .,y x nholding of course in C { x 1 , . . .,x n , y, y x 1, . . ., y x n}. We therefore readilydeduce how the derivation is transferred to the (x, a, b)-space:∂G∂y x l∂∂y x l= ∂A1 · ∂T∂y x l∂a + · · · + ∂An · ∂T1 ∂y x l∂a + ∂T∂An+1 ·n ∂y x l∂a n+1.By applying twice any two such derivations ∂ / ∂y x l 1 and ∂ / ∂y x l 2 to an arbitraryfunction G, we may see, after a few computations, what such a composeddifferentiation corresponds to, in terms of the function T defined inthe (x, a, b)-space:∂ 2 G∂y x l 1 ∂y x l 2==( n+1 ∑µ=1∑n+1µ=1∑n+1ν=1∂A µ∂y xl 1∂A µ∂y xl 1)[∂ ∑n+1∂a µ ν=1∂A ν∂y xl 2]∂A∂T ν∂y xl 2 ∂a ν∂ 2 T∂a µ ∂a + ∑n+1νµ=1∑n+1ν=1∂A µ∂y xl 1[∂ ∂A ν ] ∂T∂a µ ∂y xl 2 ∂a . νHere, by a helpful formal convention, the three Greek letters µ, ν and τ willbe used as summation indices in the total set {1, . . ., n, n+1}, while the fourLatin letters i, j, k, l will always run in the restricted set {1, . . ., n}. Replacingthen the partial derivatives of the A µ by their values (7.28) obtainedpreviously, we thus get:∂ 2 n+1G ∑=∂y x l 1 ∂y x l 2µ=1n+1∑ν=1n+1∑+n+1∑µ=1 ν=1□ µ [0 1+l1 ]□ ν [0 1+l2 ]∂ 2 T∂a µ ∂a ν +□ □{ □µ[0 1+l1 ]· □ · (∂∂a □ν µ [01+l2 ]□)− □ν[01+l2 ] · ( )∂}∂a □ µ□ · □∂T∂a ν

∂Here, the coefficients of the2 Twill not be touched anymore, but the∂a µ ∂a νcoefficients of the ∂T must be subjected to further transformations towards∂a νformal harmony, especially the numerator involving a subtraction.First of all, let us rewrite in length the concerned partial derivative of theappearing modified Jacobian determinant 28 :( ) [∂∂a □ν µ [01+l2 ] =∂□ ( a 1 | · · · |a ν−1 |0∂a µ [1+l2 ]|a ν+1 | · · · |a n+1)]= □ ( a 1 a µ | · · · |a ν−1 |0 [1+l2 ]|a ν+1 | · · · |a n+1) + · · ·++ □ ( a 1 | · · · |a ν−1 a µ |0 [1+l2 ]|a ν+1 | · · · |a n+1)+ 0++ □ ( a 1 | · · · |a ν−1 |0 [1+l2 ]|a ν+1 a µ | · · · |a n+1) + · · ·++ □ ( a 1 | · · · |a ν−1 |0 [1+l2 ]|a ν+1 | · · · |a n+1 a µ) ,and also at the same time the partial derivative of the plain Jacobiant determinant:( ) [∂∂a □ =∂□ ( a 1 | · · · |a ν−1 |a ν |a ν+1 | · · · |a n+1)]µ ∂a µ= □ ( a 1 a µ | · · · |a ν−1 |a ν |a ν+1 | · · · |a n+1) + · · ·++ □ ( a 1 | · · · |a ν−1 a µ |a ν |a ν+1 | · · · |a n+1) ++ □ ( a 1 | · · · |a ν−1 |a ν a µ |a ν+1 | · · · |a n+1) ++ □ ( a 1 | · · · |a ν−1 |a ν |a ν+1 a µ | · · · |a n+1) + · · ·++ □ ( a 1 | · · · |a ν−1 |a ν |a ν+1 | · · · |a n+1 a µ) .Consequently, the numerator with a subtraction that we want to simplifymay be rewritten in length as follows:( )∂(7.28) □ ·∂a □ν µ [01+l2 ] − □ν[01+l2 ] · ( )∂∂a □ =µ229= □ ( a 1 | · · · |a ν−1 |a ν |a ν+1 | · · · |a n+1) · □ ( a 1 a µ | · · · |a ν−1 |0 [1+l2]|a ν+1 | · · · |a n+1) + · · · +a+ □ ( a 1 | · · · |a ν−1 |a ν |a ν+1 | · · · |a n+1) · □ ( a 1 | · · · |a ν−1 a µ |0 [1+l2]|a ν+1 | · · · |a n+1) +b+ 0++ □ ( a 1 | · · · |a ν−1 |a ν |a ν+1 | · · · |a n+1) · □ ( a 1 | · · · |a ν−1 |0 [1+l2]|a ν+1 a µ | · · · |a n+1) + · · ·+c+ □ ( a 1 | · · · |a ν−1 |a ν |a ν+1 | · · · |a n+1) · □ ( a 1 | · · · |a ν−1 |0 [1+l2]|a ν+1 | · · · |a n+1 a µ) −d28 Remind that, in order to differentiate a determinant, one should differentiate separatelyeach column once and then sum all the obtained terms.

230− □ ( a 1 | · · · |a ν−1 |0 [1+l2]|a ν+1 | · · · |a n+1) · □ ( a 1 a µ | · · · |a ν−1 |a ν |a ν+1 | · · · |a n+1) a − · · · −− □ ( a 1 | · · · |a ν−1 |0 [1+l2]|a ν+1 | · · · |a n+1) · □ ( a 1 | · · · |a ν−1 a µ |a ν |a ν+1 | · · · |a n+1) b −− □ ( a 1 | · · · |a ν−1 |0 [1+l2]|a ν+1 | · · · |a n+1) · □ ( a 1 | · · · |a ν−1 |a ν a µ |a ν+1 | · · · |a n+1) OK −− □ ( a 1 | · · · |a ν−1 |0 [1+l2]|a ν+1 | · · · |a n+1) · □ ( a 1 | · · · |a ν−1 |a ν |a ν+1 a µ | · · · |a n+1) c − · · · −− □ ( a 1 | · · · |a ν−1 |0 [1+l2]|a ν+1 | · · · |a n+1) · □ ( a 1 | · · · |a ν−1 |a ν |a ν+1 | · · · |a n+1 a µ) d .The ante-penultimate underlined term “OK” will be kept untouched. To thepairs of (subtracted) □-binomials that are underlined with a, b, c, d appended(including of course all terms present in the four “· · · ”), we need anelementary instance of the Plücker identities.To state it generally, let m 2, let C 1 , C 2 , . . ., C m , D, E be (m + 2)column vectors in K m and introduce the following notation for the m ×(m + 2) matrix consisting of these vectors:[C 1 |C 2 | · · · |C m |D|E].Extracting columns from this matrix, we shall construct m×m determinantsthat are modification of the following “ground” determinant:|C 1 | · · · |C m || ≡ ∣ ∣ ∣ ∣C 1 | · · · | j 1C j1 | · · · | j 2C j2 | · · · |C m∣ ∣∣ ∣ .We use a double vertical line in the beginning and in the end to denote adeterminant. Also, we emphasize two distinct columns, the j 1 -th and thej 2 -th, where j 2 > j 1 , since we will modify them. For instance in this matrix,let us replace these two columns by the column D and by the column E,which yields the determinant:∣ ∣ ∣ C1 | · · · | j 1D| · · · | j 2E| · · · |C m∣ ∣∣ ∣.In this notation, one should understand that only the j 1 -th and the j 2 -thcolumns are distinct from the columns of the fundamental m × m “ground”determinant.Lemma. ([17], p. 155) The following quadratic identity between determinantsholds true:∣ ∣ ∣ C1 | · · · | j 1D| · · · | j 2E| · · · |C n∣ ∣∣ ∣ ·∣ ∣∣ ∣C1 | · · · | j 1C j1 | · · · | j 2C j2 | · · · |C n∣ ∣∣ ∣ == ∣ ∣ ∣ ∣C 1 | · · · | j 1D| · · · | j 2C j2 | · · · |C n∣ ∣∣ ∣ · ∣∣ ∣ ∣C 1 | · · · | j 1C j1 | · · · | j 2E| · · · |C n∣ ∣ ∣ ∣ −− ∣ ∣ ∣ ∣ C1 | · · · | j 1E| · · · | j 2C j 2| · · · |C n∣ ∣∣ ∣ ·∣ ∣ ∣ ∣C1 | · · · | j 1C j1 | · · · | j 2D| · · · |C n∣ ∣∣ ∣ .Admitting this elementary statement without redoing its proof and applyingit to all the above underlined pairs of (subtracted) monomials, afterchecking that all final signs are “−”, we obtain the following neat expression

231for (7.28):( )∂□ ·∂a □ν µ [01+l2 ] − □ν[01+l2 ] · ( )∂∂a □ = µ= −□ ( 0 [1+l2 ]| · · · | ν a ν | · · · |a n+1) · □ ( a 1 | · · · | ν a 1 a µ | · · · |a n+1) − · · · −− □ ( a 1 | · · · | ν 0 [1+l2 ]| · · · |a n+1) · □ ( a 1 | · · · | ν a ν a µ | · · · |a n+1) − · · · −− □ ( a 1 | · · · | ν a ν | · · · |0 [1+l2 ])· □(a 1 | · · · | ν a n+1 a µ | · · · |a n+1) ,or equivalently, in contracted form:( )∂□ ·∂a □ν µ [01+l2 ] − □ν[01+l2 ] · ( ) ∑n+1∂∂a □ = − µτ=1□ τ [0 1+l2 ] · □ν [a τ a µ ] .∂Thanks to this sidework, coming back to the expression for2 Gwe left∂y xl 1∂y xl 2pending above, we obtain:∂ 2 G= 1 ∑n+1∂y x l 1 ∂y x l 2 □ 2µ=1∑n+1ν=1− 1 ∑n+1□ 3µ=1{} ∂□ µ 2 T[0 1+l1 ]· □ν [0 1+l2 ]∂a µ ∂a − ν∑n+1ν=1∑n+1τ=1{} ∂T□ µ [0 1+l1 ] · □τ [0 1+l2 ] · □ ν [a µ a τ ]∂a . νTo really finalize this expression, we factor everything by1 and we exchangethe two summation indices ν and τ in the second line:□ 3{}∂ 2 G= 1 ∑n+1∑n+1□ µ ∂ 2 n+1T∂y x l 1 ∂y x l 2 □ 3 [0 1+l1 ]· □ν [0 1+l2 ] □ ·∂a µ ∂a − ∑□ τ ν [a µ a ν ] · ∂T∂a τµ=1ν=1τ=1.End of proof of the Main Theorem. As already explained in the Introduction,one applies to the system (7.28) Hachtroudi’s characterization (7.28)of equivalence to the system w ′ z ′ k 1z ′ k 2(z ′ ) = 0 with x := z, with y := w, witha := z, with b := w, with (a, b) := t, with Q := Θ, with □ := ∆, withG := Φ k1 ,k 2and with T :=∂2 Θ1∂z k1 ∂z k2. The denominator can be cleared∆ 3out, and we simply get the explicit fourth-order partial differential equationsatisfied by Θ. This completes the proof of our Main Theorem and the papermay end up now.REFERENCES[1] Bièche, C.: Le problème d’équivalence locale pour un système scalaire complet d’équationsaux dérivées partielles d’ordre deux à n variables indépendantes, Annales de la Faculté desSciences de Toulouse, XVI (2007), no. 1, 1–36.[2] Boggess, A.: CR manifolds and the tangential Cauchy-Riemann complex. Studies in AdvancedMathematics. CRC Press, Boca Raton, FL, 1991, xviii+364 pp.[3] Cartan, É.: Sur les variétés à connexion projective, Bull. Soc. Math. France 52 (1924), 205–241.

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233Lie symmetriesof partial differential equationsJoël MerkerTable of contents1. Completely integrable systems of partial differential equations . . . . . . . . . . . . . . . . . . . .??.2. Submanifold of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??.3. Classification problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??.4. Punctual and infinitesimal Lie symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??.5. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .??.6. Transfer of Lie symmetries to the parameter space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??.7. Equivalence problems and normal forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .??.8. Study of two specific examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??.9. Dual system of partial differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??.10. Fundamental pair of foliations and covering property . . . . . . . . . . . . . . . . . . . . . . . . . . . ??.11. Formal and smooth equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??.Journal of Mathematical Sciences (N. Y.), to appearThis memoir is divided in three parts 29 . Part I endeavours a general, new theory(inspired by modern CR geometry) of Lie symmetries of completely integrablePDE systems, viewed from their associated submanifold of solutions. Part II buildsgeneral combinatorial formulas for the prolongations of vector fields to jet spaces.Part III characterizes explicitly flatness of some systems of second order. The resultspresented here are original and did not appear in print elsewhere; most formulasof Parts II and III were checked by means of Maple Release 7.§1. COMPLETELY INTEGRABLE SYSTEMS OF PARTIAL DIFFERENTIALEQUATIONS1.1. General systems. Let K = R or C. Let n ∈ N with n 1 andlet x = (x 1 , . . .,x n ) ∈ K n . Also, let m ∈ N with m 1 and let y =(y 1 , . . .,y m ) ∈ K m . For α ∈ N n , we denote by a subscript y x α the partialderivative ∂ |α| y/∂x α of a local map K n ∋ x ↦→ y(x) ∈ K m .Let κ ∈ N with κ 1, let p ∈ N with p 1, choose a collection ofp multiindices β(1), . . ., β(p) ∈ N n with |β(q)| 1 for q = 1, . . .,p andmax 1qp |β(q)| = κ, and choose integers j(1), . . .,j(p) with 1 j(q) 29 Part II of [Me2005a] already appeared as [Me2005b].

234m for q = 1, . . ., p. In the present Part I, we study the Lie symmetries of ageneral system of analytic partial differential equations of the form:((E ) y j x α(x) = F αj x, y(x), ( y j(q) (x) ) ),x β(q) 1qpwhere j with 1 j m and α ∈ N n satisfy( ) ( ) ( )(1.2) j, α ≠ (j, 0) and j, α ≠ j(q), β(q) .In particular, all (κ + 1)-th partial derivatives of the unknown y = y(x)depend on a certain precise set of derivatives of order κ: the system iscomplete. In addition, all the other partial derivatives of order κ do alsodepend on the same precise set of derivatives.Here, we assume that u = 0 is a local solution of the system (E ) and thatthe functions Fα j are K-algebraic (in the sense of Nash) or K-analytic, in aneighborhood of the origin in K n+m+p . Even if our concern will be localthroughout, we will not introduce any special notation to speak of open subsetsand simply refer to various K µ . We will study five concrete instances,the first three ones being classical.Example 1.3. With n = m = κ = 1, a second order ordinary differentialequation(E 1 ) y xx = F(x, y, y x ),and more generally y x κ+1 = F(x, y, y x , . . .,y x κ), where x, y ∈ K,see [Lie1883, EL1890, Tr1896, Ca1924, Se1931, Ca1932a, Ol1986,Ar1988, BK1989, GTW1989, HK1989, Ib1992, Ol1995, N2003].Example 1.4. With n 2, m = 1 and κ = 1, a complete system of secondorder equations(E 2 ) y x i 1x i 2 = F i1 ,i 2(x i , y, y x k), 1 i1 , i 2 n,see [Ha1937, Ch1975, Su2001] and Part III below.Example 1.5. Dually, with n = 1, m 2 and κ = 1, an ordinary system ofsecond order(E 3 ) yxx j = F j( x, y j 1), y j 1x , j = 1, . . ., m,see [Fe1995, Me2004] and the references therein.Example 1.6. With n = 1, m = 2 and κ = 1, a system of the form{y2x = F ( x, y 1 , y 2 , yx)1(E 4 )y 1 xx = G( x, y 1 , y 2 , y 1 x).

Differentiating the first equation with respect to x and substituting, we getthe missing equation:(1.7)y 2 xx = F x + y 1 x F y 1 + y 2 x F y 2 + y 1 xx F y 1 x= F x + y 1 x F y 1 + y2 x F y 2 + G F y 1 x=: H ( x, y 1 , y 2 , y 1 x).Example 1.8. With n = 2, m = 1 and κ = 2, a system of the form{y x 2 = F ( )x 1 , x 2 , y, y x 1, y x(E 5 )1 x 1y x 1 x 1 x 1 = G( )x 1 , x 2 , y, y x 1, y x 1 x 1 .Here, five equations are missing. Differentiating the first equation with respectto x 1 and substituting:(1.9)y x 1 x 2 = F x 1 + y x 1 F y + y x 1 x 1 F y x 1 + y x 1 x 1 x 1 F y x 1 x 1= F x 1 + y x 1 F y + y x 1 x 1 F y x 1 + G F y x 1 x 1=: H ( x 1 , x 2 , y, y x 1, y x 1 x 1 ),and then similarly for y x 2 x 2, y x 1 x 1 x 2, y x 1 x 2 x 2, y x 2 x 2 x 2.1.10. Finitely nondegenerate generic submanifolds of C n+m . Examples1.3, 1.4, 1.6 and 1.8 (but not 1.5) are intrinsically linked to real submanifoldsof complex submanifolds.Let M be a real algebraic or analytic local generic CR 30 submanifoldof C n+m of codimension m 1 and of CR dimension n 1, and letp ∈ M. Classically, there exists local holomorphic coordinates t = (z, w) ∈C n × C m centered at p in which M is represented by(1.11) w j = Θ j (z, ¯z, ¯w), j = 1, . . .,m,for some local C-analytic map Θ = (Θ 1 , . . .,Θ m ) satisfying the identity(1.12) w ≡ Θ ( z, ¯z, Θ(¯z, z, w) ) ,reflecting the fact that M is real.Definition 1.13. ([BER1999, Me2005a, Me2005b, MP2005]) M is finitelynondegenerate if there exists an integer κ 1 such that the local holomorphicmap(1.14) (¯z, ¯w) ↦−→ ( Θ j z β(0, ¯z, ¯w)) 1jm|β|κis of rank n + m at (¯z, ¯w) = (0, 0).30 Fundamentals about Cauchy-Riemann geometry may be found in [Bo1991, BER1999,Me2005a, Me2005b, MP2005].235

236From (1.12), the map ¯w ↦→ Θ(0, 0, ¯w) is already of rank m at ¯w = 0.One then verifies ([BER1999, Me2005a, Me2005b, MP2005]) that there existmultiindices β(1), . . ., β(n) ∈ N n with |β(k)| 1 for k = 1, . . .,nand max 1kn |β(k)| = κ together with integers j(1), . . ., j(n) with 1 j(k) m such that the local holomorphic map(1.15) ( (ΘC n+m j ) ( ) )1jm,∋ (¯z, ¯w) ↦−→ (0, ¯z, ¯w) Θ j(k)z β(k)(0, ¯z, ¯w) ∈ C m+n1knis of rank n + m at (¯z, ¯w) = (0, 0).1.16. Associated system of partial differential equations. Generalizingan idea which goes back to B. Segre in [Se1931, Se1932] (n = m = 1),applied by É. Cartan in [Ca1932a] and studied more recently in [Su2001,GM2003a], we may associate to M a system of partial differential equationsof the form (E ) as follows. Complexifying the variables ¯z and ¯w, weintroduce new independent variables ζ ∈ C n and ξ ∈ C m together with thecomplex algebraic or analytic m-codimensional submanifold M of C 2(n+m)defined by(1.17) w j = Θ j (z, ζ, ξ), j = 1, . . ., m.We then consider the “dependent variables” w j as algebraic or analytic functionsof the “independent variables” z k , with additional dependence on theextra “parameters” (ζ, ξ). Then by applying the differentiation ∂ |α| /∂z αto (1.17), we get w j z α(z) = Θj zα(z, ζ, ξ). Assuming finite nondegeneracyand writing these equations for (j, α) = (j(k), β(k)), we obtain a system ofm + n equations:⎧⎨ w j (z) = Θ j (z, ζ, ξ), j = 1, . . ., m,(1.18)⎩w j(k)(z) = Θ j(k)z β(k) zβ(k)(z, ζ, ξ),k = 1, . . .,n.By means of the implicit function theorem we can solve:(1.19) (ζ, ξ) = R ( z k , w j (z), w j(k)z β(k) (z) ) .Finally, for every pair (j, α) different from (j, 0) and from (j(k), β(k)),we may replace (ζ, ξ) by R in the differentiated expression w j z α(z) =Θ j zα(z, ζ, ξ), which yields(w j z α(z) = Θj z z, R ( z k , w j (z), w j(k) (z) ))α z(1.20)(β(k) )=: Fαj z k , w j (z), w j(k) (z) .z β(k)This is the system of partial differential equations associated to M.

Example 1.21. (Continued) With n = m = 1, i.e. M ⊂ C 2 and κ = 1, i.e.M is Levi nondegenerate of equation(1.22) w = ¯w + i z¯z + O 3 ,where z, ¯z are assigned weight 1 and w, ¯w weight 2, B. Segre [Se1931]obtained w zz = F(z, w, w z ). J. Faran [Fa1980] found some examples ofsuch equations that cannot come from a M ⊂ C 2 . But the following wasleft unsolved.Open problem 1.23. Characterize equations y xx = F(x, y, y x ) associated toa real analytic, Levi nondegenerate (i.e. κ = 1) hypersurface M ⊂ C 2 . Canon read the reality condition (1.12) on F ? In case of success, generalize toarbitrary M ⊂ C n+m .Example 1.24. (Continued) Similarly, the system (E 2 ) comes from a Levinondegenerate hypersurface M ⊂ C n+1 ([Ha1937, CM1974, Ch1975,Su2001]. Exercise: why (E 3 ) cannot come from any M ⊂ C ν ?Example 1.25. (Continued) With n = 1, m = 2 and κ = 1, the system (E 4 )comes from a M ⊂ C 3 which is Levi nondegenerate and satisfies(1.26) T c M + [T c M, T c M] + [ T c M, [T c M, T c M] ] = TMat the origin, namely which has equations of the following form, after someelementary transformations ([Be1997, BES2005]):(1.27)w 1 = ¯w 1 + i z¯z + O 4 ,w 2 = ¯w 2 + i z¯z(z + ¯z) + O 4 ,where z, ¯z are assigned weight 1 and w 1 , w 2 , ¯w 1 , ¯w 2 weight 2.Example 1.28. (Continued) With n = 2, m = 1 and κ = 2, the system (E 5 )comes from a hypersurface M ⊂ C 3 of equation ([Eb1998, GM2003b,FK2005a, FK2005b, Eb2006, GM2006]):(1.29) w = ¯w + i 2 z1¯z 1 + z 1 z 1¯z 2 + ¯z 1¯z 1 z 21 − z 2¯z 2 + O 4 ,where z 1 , ¯z 1 , z 2 , ¯z 2 are assigned weight 1 and w, ¯w weight 2, with theassumption that the Levi form has rank exactly one at every point, and withthe assumption that M is 2-nondegenerate at 0.1.30. Jet spaces, contact forms and Frobenius integrability. Throughoutthe present Part I, we assume that the system (E ) is completely integrable,namely that the Pfaffian system naturally associated to (E ) in the appropriatejet space is involutive in the sense of Frobenius. This holds automatically incase (E ) comes from a generic submanifold M ⊂ C n+m . In general, we willconstruct a submanifold of solutions associated to (E ). So, we must explaincomplete integrability.237

238We denote by Jn,m κ the space of κ-th jets of maps K n ∋ x ↦→ y(x) ∈ K m .Let(1.31)(x i , y j , y j i 1, y j i 1 ,i 2, . . .. . .,y j i 1 ,i 2 ,...,i κ)∈ Kn+m+mn+mn 2 +···+mn κ ,denote the natural coordinates on Jn,m κ ≃ Kn+m(1+n+···+nκ) . For instance,(x, y, y 1 ) ∈ J1,1 1 . We shall sometimes write them shortly:(1.32)(x i , y j , y j )β ∈ Kn+m+m(n+···+n κ) ,where β ∈ N n varies and satisfies |β| κ. Sometimes also, weconsider these jet coordinates only up to their symmetries y j i 1 ,i 2 ,...,i λ=y j i σ(1) ,i σ(2) ,...,i σ(λ), where σ is a permutation of {1, 2, . . ., λ}, so that J κ n,m ≃K n+m Cκ n+κ , with Cκn+κ := (n+κ)! .κ! n!Having these notations at hand, we may develope the canonical system ofcontact forms on Jn,m κ ([Ol1995], [Stk2000]):⎧n∑θ j := dy j − y j k dxk ,(1.33)⎪⎨θ j i 1:= dy j i 1−k=1n∑y j i 1 ,k dxk ,k=1· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·n∑⎪⎩ θ j i 1 ,...,i κ−1:= dy j i 1 ,...,i κ−1− y j i 1 ,...,i κ−1 ,k dxk .For instance, with n = m = 1 and κ = 2, we have θ 1 = dy −y 1 dx and θ1 1 =dy 1 − y 2 dx. These (linearly independent) one-forms generate a subspaceC T κ n,m of the cotangent T ∗ Jn,m κ whose dimension equals m Cκ−1 n+κ−1. Forthe duality between forms and vectors, the orthogonal (C T κ n,m )⊥ in TJn,mκis spanned by the n + m Cn+κ−1 κ vector fields:(1.34) ⎧⎪⎨⎪⎩D i := ∂∂x i + m∑T j 1i 1 ,...,i κ:=j 1 =1∂∂y j ,1i 1 ,...,i κy j 1ik=1∂∂y + · · · + ∑ m j 1n∑j 1 =1 k 1 ,...,k κ−1 =1y j 1i,k 1 ,...,k κ−1∂∂y j ,1k 1 ,...,k κ−1the first n ones being the total differentiation operators, considered in Part II.For n = m = 1, κ = 2, we get ∂ + y ∂x 1 ∂ + y ∂y 2 ∂∂y 1and ∂∂y 2.Classically ([Ol1986, BK1989, Ol1995]), one associates to (E ) its skeleton∆ E , namely the (n + m + p)-dimensional submanifold of Jn,m κ+1 simply

defined by the graphed equations:((1.35) yα j = Fαj x, y, ( y j(q) )β(q)1qpfor ( j, α ) ≠ (j, 0) and ≠ ( j(q), β(q) ) with |α| κ + 1. Clearly, the naturalcoordinates on ∆ E are:(1.36) (x, y, ( y j(q)β(q))1qp)≡),(x, y, ( y j(q) ) )l 1 (q),...,l λq (q)∈ K n × K m × K p ,1qpwhere λ q := |β(q)| and ( l 1 (q), . . ., l λq (q) ) := β(q).Next, in view of the form (1.34) of the generators of (C T κ+1n,m )⊥ and inview of the equations of ∆ E , the intersection(1.37) (C T κ+1n,m )⊥ ∩ T∆ Eis a vector subbundle of T∆ E that is generated by n linearly independentvector fields obtained by restricting the D i to ∆ E , which yields:⎧D i = ∂ ∑ m ( )⎪⎨ ∂x i+ A j i x i 1, y j 1, y j(q 1) ∂β(q 1 )∂y + j j=1(1.38)⎪⎩+p∑q=1B q i(x i 1, y j 1, y j(q 1)β(q 1 )) ∂∂y j(q)β(q)i = 1, . . .,n, where the coefficients A j i and Bq i are given by:(1.39) {yjA j i := i if the variable yj i appears among the p variables yj(q 1)β(q 1 ) ;F ji otherwise;B q i := ⎧⎨⎩,239y j(q)i,l 1 (q),...,l λq (q)if yj(q)l 1 (q),...,l λq(q)appears among the p variables y j(q 1)β(q 1 ) ;F j(q)i,l 1 (q),...,l λq (q) otherwise.Example 1.40. For (E 1 ), we get D = ∂ + y ∂x 1 ∂ + F(x, y, y ∂x 1) ∂∂y 2; exercise:treat (E 2 ) and (E 3 ). For (E 4 ), we get D = ∂ + ∂x y1 1 ∂ + F ∂ + G ∂ . For∂y 1 ∂y 2 ∂y11(E 5 ), whose skeleton is written y 2 = F , y 1,1,1 = G, y 1,2 = H, y 1,1,2 = K,with F , G, H, K being functions of ( )x 1 , x 2 , y, y 1 , y 1,1 , we get(1.41)D 1 = ∂∂x 1 + y 1∂∂y + y 1,1∂+ G ∂ ,∂y 1 ∂y 1,1D 2 = ∂∂x 2 + F ∂ ∂y + H ∂∂y 1+ K∂∂y 1,1.Definition 1.42. The system (E ) is completely integrable if the n vectorfields (1.38) satisfy the Frobenius integrability condition, namely every Lie

240bracket [D i1 , D i2 ], 1 i 1 , i 2 n, is a linear combination of the vector fieldsD 1 , . . .,D n .Because of their specific form (1.38), we must then have in fact[D i1 , D i2 ] = 0. For n = 1, the condition is of course void.§2. SUBMANIFOLD OF SOLUTIONS2.1. Fundamental foliation of the skeleton. As the vector fields D i commute,they equip the skeleton ∆ E ≃ K n+m+p with a foliation F ∆E by n-dimensional integral manifolds which are (approximately) directed alongthe x-axis. We draw a diagram (see only the left side).y xyD0abxyDM (E)0a, bxDDDexp(xD)(0,a, b)The (abstract, not numerical) integration of (E ) is thus straightforwardlycompleted: the set of solutions coincides with the set of leaves of F ∆E . Thisis the true geometric content, viewed in the appropriate jet space, of theassumption of complete integrability.2.2. General solution and submanifold of solutions. To construct the submanifoldof solutions M (E ) associated to (E ) (sketched in the right handside), we execute some elementary analytico-geometric constructions.At first, we duplicate the coordinates ( y j(q)β(q) , yj) ∈ K p × K m by introducinga new subspace of coordinates (a, b) ∈ K p × K m ; thus, on the leftdiagram, we draw a vertical plane together with a- and b-axes. The leavesof the foliation F ∆E are uniquely determined by their intersections with thisplane, consisting of points of coordinates (0, a, b) ∈ K n × K p × K m .Such points (0, a, b) correspond to the initial conditions ( y j(q)x β(q) (0), y(0) )for the general solution of (E ). In fact, the (concatenated, multiple) flow of{D 1 , . . .,D n } is given by(2.3)exp ( x n D n(· · ·(exp(x 1 D 1 (0, a, b))) · · ·)) = ( x, Π(x, a, b), Ω(x, a, b) ) ∈ K n ×K m ×K p ,

for some two local analytic maps Π = (Π 1 , . . .,Π m ) and Ω = (Ω 1 , . . .,Ω p )and the next lemma is straightforward.Lemma 2.4. The general solution of (E ) is(2.5) y(x) := Π(x, a, b),where (a, b) varies in K p × K m . Furthermore, for q = 1, . . .,p:(2.6) Ω q (x, a, b) ≡ Π j(q)x β(q) (x, a, b).This leads to introducing a fundamental geometric object.Definition 2.7. The submanifold of solutions V S (E ) associated to (E ) isthe analytic submanifold of K n x × K m y × K p a × K m b defined by the Cartesianequations:(2.8) 0 = −y j + Π j (x, a, b), j = 1, . . .,m.There is a strong interplay between the study of (E ) and the geometry ofV S (E ). By construction, the diffeomorphism:(2.9)⎧(⎨ A : K n+p+m [coordinates (x i , a q , b j )] −→ K[coordinatesn+m+p()⎩A (x i , a q , b j ) := x i , Π j (x, a, b), Π j(q) (x, a, b), ,x β(q)sends the foliation F v by the variables x whose leaves are {a = cst., b =cst.} (see the diagram), to the previous foliation F ∆E .2.10. PDE system associated to a submanifold. Inversely, let M be a submanifoldof K n x × Km y × Kp a × Km b of the form(2.11) y j = Π j (x, a, b), j = 1, . . ., m.A necessary condition for it to be the complexification of a generic M ⊂C n+m is that p = n (answer to an exercise above).241x i , y j , y j(q)β(q)Definition 2.12. M is solvable with respect to the parameters ifb ↦→ Π(0, 0, b) of rank m at b = 0 and if there exist κ 1, multiindicesβ(1), . . ., β(p) ∈ N n with |β(q)| 1 for q = 1, . . .,pand max 1qp |β(q)| = κ, together with integers j(1), . . ., j(p) with1 j(q) m such that the local K-analytic map(2.13) ( (ΠK m+p ∋ (a, b) ↦−→ j (0, a, b) ) ( ) )1jm, Π j(q) (0, a, b) ∈ K m+px β(q) 1qpis of rank equal to m + p at (a, b) = (0, 0))]

242When M is the submanifold of solutions of a system (E ), it is automaticallysolvable with respect to the variables, the pairs (j(q), β(q)) being thesame as in the arguments of the right hand sides Fα j in (E ). Proceeding asin §1.16, we may associate to M a system of the form (E ). Since we needintroduce some new notation, let us repeat the argument.Considering y = y(x) = Π(x, a, b) as a function of x with extra parameters(a, b) and applying ∂ |α|/ ∂x α , we get y j x α(x) = Πj xα(x, a, b). Writingonly the relevant (m + p) equations:{y j (x) = Π j (x, a, b),(2.14)y j(q) = Π j(q) (x, a, b),x β(q) x β(q)the assumption of solvability with respect to parameters enables to get(2.15)⎧⎨⎩a q = A q( x i , y j , y j(q 1)β(q 1 )),b j = B j( x i , y j 1, y j(q)β(q)).For every (j, α) ≠ (j, 0) and ≠ (j(q), β(q)), we then replace (a, b) in y j x = αΠ j x α:(y j x(2.16)α(x) = Πj x x, A ( x i , y j 1(x), y j(q)α β(q) (x)) , B ( x i , y j 1(x), y j(q)β(q) (x)))(=: Fαj x i , y j 1(x), y j(q) (x) ) .x β(q)Proposition 2.17. There is a one-to-one correspondence(2.18) (E M ) = (E ) ←→ M = M (E ) ,between completely integrable systems of partial differential equations ofthe general form (E ) and submanifolds (of solutions) M of the form (2.11)which are solvable with respect to the parameters. Of course( )(2.19) EM(E) = (E ) and M(EM ) = M.2.20. Transfer of total differentiations. We notice that the auxiliary functionsA q and B j enable to express the inverse of A:(2.21)A −1 :(xi 1, y j 1, y j(q )1)β(q 1 ) ↦−→(x i , A q( x i 1, y j 1, y j(q )1)(β(q 1 ) , Bjx i 1, y j 1, y j(q 1)β(q 1 )) ) .More importantly, the total differentiation operator considerably simplifieswhen viewed on M . This observation is useful for translating differentialinvariants of (E ) as differential invariants of M .Lemma 2.22. Through A, for i = 1, . . .,n, the pull-back of the total differentiationoperator D i is simply ∂ , or equivalently:∂x( i ∂)(2.23) A ∗ = D∂x i i .

Proof. Let l = l ( x i , y j , yβ(q)) j(q) be any function defined on ∆E . Composingwith A yields the function Λ := l ◦ A, i.e.((2.24) Λ(x, a, b) ≡ lx i , Π j (x, a, b), Π j(q)x β(q) (x, a, b)Differentiating with respect to x i , we get, dropping the arguments:(2.25)∂Λ∂x = ∂li ∂x + ∑ m ij=1Π j x i∂lp∑∂y + jq=1Π j(q)x i x β(q)).∂l.∂y j(q)x β(q)Replacing the appearing Π j xα for which (j, α) ≠ (j, 0) and ≠ (j(q), β(q))by Fα j, we recover D i as defined by (1.38), whence ∂Λ = D∂x i i l.2.26. Transfer of algebrico-differential expressions. The diffeomorphismA may be used to translate algebrico-differential expressions from M to (E )and vice-versa:(2.27) I M(Jλ+κ+1x,a,bΠ ) ←→ I (E )(Jλx,y,y1 F ) .Here, λ ∈ N, the letter J is used to denote jets, and I = I M or = I (E) is apolynomial or more generally, a quotient of polynomials with respect to itsjet arguments. Notice the shift by κ + 1 of the jet orders.Example 2.28. Suppose n = m = 1 and κ = 1. Then F = Π xx . As anexercise, let us compute F x , F y , F y1 in terms of Jx,a,b 3 Π. We start with theidentity(2.29) F(x, y, y 1 ) ≡ Π xx(x, A(x, y, y1 ), B(x, y, y 1 ) ) ,that we differentiate with respect to x, to y and to y 1 :F x = Π xxx + Π xxa A x + Π xxb B x ,(2.30)F y = Π xxa A y + Π xxb B y ,F y1 = Π xxa A y1 + Π xxb B y1 .Thus, we need to compute A x , A y , A y1 , B x , B y , B y1 . This is easy: it sufficesto differentiate the two identities that define A and B as implicit functions,namely:y ≡ Π ( x, A(x, y, y 1 ), B(x, y, y 1 ) )and(2.31)(y 1 ≡ Π x x, A(x, y, y1 ), B(x, y, y 1 ) )with respect to x, to y and to y 1 , which gives six new identities:0 = Π x + Π a A x + Π b B x , 0 = Π xx + Π xa A x + Π xb B x ,(2.32)1 = Π a A y + Π b B y , 0 = Π xa A y + Π xb B y0 = Π a A y1 + Π b B y1 , 1 = Π xa A y1 + Π xb B y1 ,and to solve each of the three linear systems of two equations located ina line, noticing that their common determinant Π b Π xa − Π a Π xb does not243

244vanish at the origin, since Π = b+xa+O 3 . By elementary Cramer formulas,we get:⎧A x = −Π b Π xx + Π x Π xb, B x = −Π x Π xa + Π a Π xx,Π b Π xa − Π a Π xb Π b Π xa − Π a Π xb(2.33)⎪⎨⎪⎩ A y1 =−Π xbA y =,Π b Π xa − Π a Π xbB y =Π b,Π b Π xa − Π a Π xbB y1 =Π xaΠ b Π xa − Π a Π xb,−Π aΠ b Π xa − Π a Π xb.Replacing in (2.30), no simplification occurs and we get what we wanted:(2.34)⎧F x = Π xxx + Π [ ] [ ]xxa − Πb Π xx + Π x Π xb + Πxxb − Πx Π xa + Π a Π xx,Π ⎪⎨b Π xa − Π a Π xbF y = −Π xxa Π xb + Π xxb Π xa,Π b Π xa − Π a Π xb⎪⎩ F y1 = Π xxa Π b − Π xxb Π a,Π b Π xa − Π a Π xbOne sees D F = F x + Π x F y + Π xx F y1 = Π xxx simply, as predicted byLemma 2.22.Second order derivatives F xx , F xy , F xy1 , F yy , F yy1 , F y1 y 1have still reasonablecomplexity, when expressed in terms of Jx,a,b 4 Π. Beyond, the computationsexplode.Open question 2.35. A second order ordinary differential equation y xx =F(x, y, y x ) has two fundamental differential invariants, namely ([Tr1896,Ca1924, GTW1989, Ol1995]):(2.36)I 1 (E 1 ) := ∂4 Fand∂y14I 2 (E 1 ) := DD ( )F y1 y 1 − Fy1 D ( ) ( )F y1 y 1 − 4 D Fyy1 + 6 Fyy − 3 F y F y1 y 1+ 4 F y1 F yy1 .Compute I 1 M 1and I 2 M 1.Although the notion of diffeomorphism is clear and apparently obviousfrom the intuitive, geometric and conceptual viewpoints, in concrete applicationsand in explicit computations, it almost never straightforward totransfer algebrico-differential objects.Open problem 2.37. For general (E ) and M , build closed combinatorialformulas executing the double translation (2.27).2.38. Plan for the sequel. We will endeavour a general theory showing thatthe study of systems (E ) and the study of submanifolds of solutions M gives

complementary views on the same object. In fact, Lie symmetries, equivalenceproblems, Cartan connections, normal forms and classification listsmay be endeavoured on both sides, yielding essentially equivalent results,though the translation is seldom straightforward. In Section 3, 4 and 5, wereview some features from the side (E ), before studying some aspects fromthe side of M . A more systematic and complete approach shall appear as amonography.245§3. CLASSIFICATION PROBLEMS3.1. Transformations of PDE systems. Through a local K-analytic changeof variables close to the identity (x, y) ↦→ ϕ(x, y) =: (x ′ , y ′ ), the system (E )transforms to a similar system, with primes:((E ′ ) y ′ jx ′α(x′ ) = F ′ jα x ′ , y ′ (x ′ ), ( y ′ j(q)(x ′ ) ) ).x ′β(q) 1qpExample 3.2. Coming back temporarily to the notations of §1.12(II), withn = m = κ = 1, assume that y xx = f(x, y, y x ) transforms to( Y XX = F(X, Y, Y) X ) through a local diffeomorphism (x, y) ↦→ (X, Y ) =X(x, y), Y (x, y) . How F is related to f ? By symmetry, it suffices tocompute f in terms of F , X, Y . The prolongation to J1,1 2 of the diffeomorphismhas components ([BK1989, Me2004]):(3.3) Y X = Y x + y x Y yX x + y x X y,and(3.4)1Y XX = [ ] 3(y xx ·∣ X ∣ x X y ∣∣∣ +Xx + y x X yY x Y y∣ X ∣x X xx ∣∣∣+Y x Y xx{+y x · 2∣ X ∣ x X xy ∣∣∣ −Y x Y xy∣ X ∣}xx X y ∣∣∣+Y xx Y y{∣ ∣ ∣∣∣ X+y x y x · x X yy ∣∣∣ − 2Y x Y yy∣ X ∣}xy X y ∣∣∣+Y xy Y y{+y x y x y x · −∣ X ∣})yy X y ∣∣∣.Y yy Y y

246It then suffices to replace Y XX above by F(X, Y, Y X ) and to solve y xx :(3.5) ( (1 [Xx ]y xx =∣ X ∣ 3x X y ∣∣∣+ y x X y F X, Y, Y )x + y x Y y−X x + y x X y∣ X ∣x X xx ∣∣∣+Y x Y xxY x Y y{+y x · −2∣ X ∣ x X xy ∣∣∣ +Y x Y xy∣ X ∣}xx X y ∣∣∣+Y xx Y y{+y x y x · −∣ X ∣ x X yy ∣∣∣ + 2Y x Y yy∣ X ∣}xy X y ∣∣∣+Y xy Y y{∣ ∣})∣∣∣ X+y x y x y x · yy X y ∣∣∣Y yy Y y=: f(x, y, y x ).Open problem 3.6. Find general formulas expressing the F j α in terms of F ′ jα,x ′i , y ′j .Conversely, given two such systems (E ) and (E ′ ), when do they transformto each other ? Let π κ,p ′ denote the projection from J ′ κ+1n,m to ∆ E ′ definedby((3.7) π κ,p′ x ′i , y ′j , y ′ ji 1, . . .,y ′ j) ( )i 1 ,...,i κ+1 := x ′i , y ′j , y ′ j(q).β(q)Let ϕ (κ+1) be the (κ + 1)-th prolongation of ϕ (Section 1(II)).Lemma 3.8. ([Ol1986, BK1989, Ol1995]) The following three conditionsare equivalent:(1) ϕ transforms (E ) to (E ′ );(2) its (κ+1)-th prolongation ϕ (κ+1) : Jn,m κ+1 → J ′ κ+1n,m maps ∆ E to ∆ E ′;(3) ϕ (κ+1) : Jn,m κ+1 → J ′ κ+1n,m maps ∆ E to ∆ E ′ and the associated map(3.9) Φ E ,E ′ := π κ,p ′ ◦ ( ϕ (κ+1)∣ )∣∆Esends every leaf of F ∆E to some leaf of F ∆E ′.Equivalence problem 3.10. Find an algorithm to decide whether two given(E ) and (E ′ ) are equivalent.Élie Cartan’s widely applicable method (not reviewed here; [Ca1937,Ste1983, G1989, HK1989, Fe1995, Ol1995]) provides an answer “in principle”to this question by reducing to an {e}-structure an initial G-structureassociated to (E ). Due to the incredible size-length-complexity of the underlyingcomputations, this approach almost never abutes: it is forced toincompleteness. But in fact, the main question is to classify.

Classification problem 3.11. Classify systems (E ), namely provide completelists of all possible such equations written in simplified “normal”, easilyrecognizable forms.Both problems are deeply linked to the classification of Lie algebras oflocal vector fields. For n = 1, m = 1 and κ = 1, namely (E 1 ): y xx =F(x, y, y x ), Lie and Tresse solved the two problems 31 . Table 7 of [Ol1986],below reproduced, describes the results.247Symmetry group Dimension Invariant equation(1) 0 y xx = F(x, y, y x )(2) ∂ y 1 y xx = F(x, y x )(3) ∂ x , ∂ y 2 y xx = F(y x )(4) ∂ x , e x ∂ y 2 y xx − y x = F(y x − y)(5) ∂ x , ∂ x − y∂ y , x 2 ∂ x − 2xy∂ y 3 y xx = 3y2 x2y cy3(6) ∂ x , x∂ x − y∂ y , 3 y xx = 6yy x − 4y 3 +x 2 ∂ x − (2xy + 1)∂ y +c(y x − y 2 ) 3/2(7) ∂ x , ∂ y , x∂ x + αy∂ y , 3 y xx = c(y x ) α−2α−1α ≠ 0, 1, 1, 2 2(8) ∂ x , ∂ y , x∂ x + (x + y)∂ y 3 y xx = ce −yx(9) ∂ x , ∂ y , y∂ x , x∂ y , y∂ y , 8 y xx = 0x 2 ∂ x + xy∂ y , xy∂ x + y 2 ∂ yTable 1.However, the author knows no modern reference offering a completeproof of this classification, with precise insight on the assumptions (somenormal forms hold true only at a generic point). In addition, the above Lie-Tresse list is still slightly incomplete in the sense that it does not precisewhich are the conditions satisfied by F (Table 7 in [Ol1986]) insuring in thefirst four lines that SYM(E 1 ) is indeed of small dimension 0, 1 or 2.Open question 3.12. Specify some precise nondegeneracy conditions uponF in the first four lines of Table 1.§4. PUNCTUAL AND INFINITESIMAL LIE SYMMETRIES4.1. Lie symmetries of (E ). Let ϕ = (φ, ψ) be a diffeomorphism of K n x ×K n y as in (1.7)(II).31 The author knows no complete confirmation of the Lie-Tresse classification by meansof É. Cartan’s method of equivalence.

248Definition 4.2. ([Ol1986, Ol1995, BK1989]) ϕ is a (local) Lie symmetryof (E ) if it transforms the graph of every solution of (E ) into the graph ofanother solution.To explain, we must pass to jet spaces. Denote the components of the(κ + 1)-th prolongation ϕ (κ+1) : Jn,m κ+1 → Jn,m κ+1 by(4.3) ϕ (κ+1) = ( φ i 1, ψ j 1, Φ j i 1, Φ j i 1 ,i 2, . . .. . . , Φ j i 1 ,i 2 ,...,i κ+1).The restriction ϕ (κ+1)∣ ∣∆Eis obtained by replacing each jet variable yα j byFα j , whenever (j, α) ≠ (j, 0) and ≠ (j(q), β(q)), and wherever it appears32in the Φ j i 1 ,...,i λ.Let π κ,p denote the projection from Jn,m κ+1 to ∆ E ≃ K m+n+p defined by((4.4) π κ,p x i , y j , y j i 1, . . .,y j ) ( )i 1 ,...,i κ+1 := x i , y j , y j(q) ,and introduce the map(4.5) ϕ ∆E := π κ,p ◦ ( ϕ (κ+1)∣ ∣∆E)≡(ϕ(x i , y j ), Φ j(q)β(q)β(q)(x i , y j , y j(q 1)β(q 1 )) ) .Lemma 4.6. ([Ol1986, Ol1995, BK1989], [∗]) The following three conditionsare equivalent:(1) the diffeomorphism ϕ is a Lie symmetry of (E );(2) ϕ (κ+1)∣ ∣∆Esends ∆ E to ∆ E ;(3) ϕ (κ+1)∣ ∣∆Esends ∆ E to ∆ E and ϕ ∆E = π κ,p(ϕ(κ+1) ∣ ∣∆E)is a symmetryof the foliation F ∆E , namely it sends every leaf to some other leaf.Then the set of Lie symmetries of (E ) constitutes a local Lie(pseudo)group.4.7. Infinitesimal Lie symmetries of (E ). Letn∑(4.8) L = X i (x, y) ∂∂x + ∑ mY j (x, y) ∂i ∂y , ji=1be a (local) vector field on K n+m having analytic coefficients. Denote itsflow by ϕ t (x, y) := exp(t L )(x, y), t ∈ K. As in Section 1(II), by differentiatingthe prolongation (ϕ t ) (κ+1) with respect to t at t = 0, we get theprolonged vector field L (κ+1) on Jn,m κ+1 , having the general form (Part II):(4.9)L (κ+1) = L +m∑j=1n∑i 1 =1Y j i 1∂∂y j i 1+· · ·+j=1m∑j=1with known explicit expressions for the Y j i 1 ,...,i λ.n∑i 1 ,...,i κ+1 =1Y j i 1 ,...,i κ+1∂∂y j i 1 ,...,i κ+1,32 Remind from Section 1(II) that we have not (open problem) provided a complete explicitexpression of Φ j i 1,...,i λfor general n 1, m 1 and λ 1.

Definition 4.10. L is an infinitesimal symmetry of (E ) if for every small t,its time-t flow map ϕ t is a Lie symmetry of (E ).The restriction L (κ+1)∣ ∣∆Eis obtained by replacing every yα j by F α j in allcoefficients Y j i 1, . . .,Y j i 1 ,...,i κ+1. Then the coefficients become functions of(xi 1, y j 1, y j(q 1)β(q 1 ))only.Lemma 4.11. ([Ol1986, Ol1995, BK1989], [∗]) The following three conditionsare equivalent:(1) the vector field L is an infinitesimal Lie symmetry of (E );(2) its (κ + 1)-th prolongation L (κ+1) is tangent to the skeleton ∆ E ;(3) L (κ+1) is tangent to ∆ E and the push-forward(4.12) L ∆E := (π κ,p ) ∗(L(κ+1) ∣ ∣∆E)249is an infinitesimal symmetry of the foliation F ∆E , namely for everyi = 1, . . ., n, the Lie bracket [ L ∆E , D i]is a linear combination of{D 1 , . . .,D n }.According to [Ol1986, BK1989, Ol1995], the set of infinitesimal Lie symmetriesconstitutes a Lie algebra, with the property [ L (κ+1) , L ′ (κ+1) ] =[L , L′ ] (κ+1) . We summarize by a diagram.ϕ (κ+1)Jn,m κ+1Jn,mκ+1L (κ+1)∆ Eπ κ,pϕ ∆Eπ κ,p∆ Eπ κL ∆Eπ κπ pπ pK n x × K m yϕK n x × K m yL4.13. Sophus Lie’s algorithm. We describe the general process. Its complexitywill be exemplified in Section 5 (to be read simultaneously).The tangency of L (κ+1) to ∆ E is expressed by applying L (κ+1) to theequations 0 = −yα j + F α j , which yields:(4.14) 0 = −Y j α +n∑i=1X i ∂F j α∂x i +n∑l=1Y l ∂F j α∂y l +p∑q=1Y j(q)β(q)∂Fαj ,∂y j(q)β(q)for (j, α) ≠ (j, 0) and ≠ (j(q), β(q)). Restricting a coefficient Y j i 1 ,...,i λto∆ E , namely replacing everywhere in it each yα j by F α j , provides a specialized

250coefficient(4.15) Ŷ j i 1 ,...,i λ= Ŷj i 1 ,...,i λ(xi 1, y j 1, y j(q 1)β(q 1 ) , Jλ x,yX i 1, J λ x,yY j 1 ) ,that depends linearly on the λ-th jet of the coefficients of L , as confirmedby an inspection of Part II’s formulas. Here, we use the jet notationJx,y λ Z := ( ∂ α 1x ∂β 1y Z ) |α 1 |+|β 1. We thus get equations|λn∑(4.16) 0 ≡ −Ŷj α + X i ∂F αj n∑+ Y l ∂F j p∑α+ Ŷ j(q) ∂Fαj∂x i ∂y l β(q),i=1l=1q=1∂y j(q)β(q)involving only the variables ( x i 1, y j 1, y j(q 1)β(q 1 )).Next, we develope every such equation with respect to the powers ofy j(q 1)β(q 1 ) :(4.17)0 ≡ ∑µ 1 ,...,µ p0(y j(1)β(1) )µ1 · · ·(y j(p)β(p) )µp Ψ j α,µ 1 ,...,µ p(xi 1, y j 1, J κ+1x,y X i 1, J κ+1x,y Y j 1 ) .The Ψ j α,µ 1 ,...,µ pare linear with respect to ( J κ+1x,y X i 1, J κ+1x,y Y j 1), with certaincoefficients analytic with respect to (x, y), which depend intrinsically (butin a complex manner) on the right hand sides F j α.Proposition 4.18. The vector field L is an infinitesimal Lie symmetry of(E ) if and only if its coefficients X i 1, Y j 1satisfy the linear PDE system:((4.19) 0 = Ψ j α,µ 1 ,...,µ p xi 1, y j 1, Jx,y κ+1 X i 1, Jx,y κ+1 Y ) j 1for all (j, α) ≠ (j, 0) and ≠ (j(q), β(q)) and for all (µ 1 , . . .,µ p ) ∈ N p .In all known instances, a finite number of these equations suffices.Example 4.20. With n = m = κ = 1, a second prolongation L (2) =X ∂ + Y ∂ + Y ∂x ∂y 1 ∂ ∂∂y 1+ Y 2 ∂y 2is tangent to the skeleton 0 = −y 2 +F(x, y, y 1 ) of (E 1 ) if and only if 0 = −Y 2 + X F x + Y F y + Y 1 F y1 , or,developing:(4.21) ⎧⎪⎨0 = −Y xx + [ ]− 2 Y xy + X xx y1 + [ ]− Y yy + 2 X xy (y1 ) 2 + [ ]X yy (y1 ) 3 ++ [ ] [ ]− Y y + 2 X x F + 3 Xy y1 F + [ X ] F x + [ Y ] F y +⎪⎩+ [ ]Y x Fy1 + [ ]Y y − X x y1 F y1 + [ ]− X y (y1 ) 2 F y1 .Developing F = ∑ k0 (y 1) k F k (x, y), we may obtain equations (4.19).§5. EXAMPLES5.1. Second order ordinary differential equation. Pursuing the study of(E 1 ), according to Section 7 below, we may assume that F = O(y x ), orequivalently F(x, y, 0) ≡ 0.

Convention 5.2. The letters R will denote various functions of (x, y, y 1 ),changing with the context. Similarly, r = r(x, y), excluding the pure jetvariable y 1 . Hence, symbolically:(5.3) R = r + y 1 r + (y 1 ) 2 r + (y 1 ) 3 r + · · · .So the skeleton is(5.4) y 2 = F(x, y, y 1 ) = y 1 R = y 1 r + (y 1 ) 2 r + (y 1 ) 3 r + · · · .Applying L (2) , see (2.3)(II) for its expression, we get:(5.5) 0 = −Y 2 + X F x + Y F y + Y 1 F y1 .Observe that F x = (y 1 R) x = r y 1 +r (y 1 ) 2 +· · · and similarly for F y , but that(y 1 R) y1 = r +r y 1 +r (y 1 ) 2 + · · · . Inserting above Y 1 , Y 2 given by (2.6)(II),replacing y 2 by y 1 R and computing mod (y 1 ) 4 , we get:(5.6)0 ≡ − Y xx + [ − 2 Y xy + X xx]y1 + [ − Y yy + 2 X xy](y1 ) 2 + [ X yy](y1 ) 3 ++ [ − Y y + 2 X x](y1 r + (y 1 ) 2 r + (y 1 ) 3 r ) + [ 3 X y] ((y1 ) 2 r + (y 1 ) 3 r ) ++ [ X ]( y 1 r + (y 1 ) 2 r + (y 1 ) 3 r ) + [ Y ]( y 1 r + (y 1 ) 2 r + (y 1 ) 3 r ) ++ [ Y x] (r + y1 r + (y 1 ) 2 r + (y 1 ) 3 r ) ++ [ Y y − X x](y1 r + (y 1 ) 2 r + (y 1 ) 3 r ) + [ − X y]((y1 ) 2 r + (y 1 ) 3 r ) .We gather the powers cst., y 1 , (y 1 ) 2 and (y 1 ) 3 , equating their coefficients to0:(5.7)0 = −Y xx + P ( Y x),0 = −2 Y xy + X xx + P ( Y y , X x , X , Y , Y x),0 = −Y yy + 2 X xy + P ( Y y , X x , X y , X , Y , Y x),0 = X yy + P ( Y y , X x , X y , X , Y , Y x)Convention 5.8. The letter P will denote various linear combinations ofsome precise partial derivatives of X , Y which have analytic coefficientsin (x, y).By cross-differentiations and substitutions in the above system, allthird, fourth, fifth, etc. order derivatives of X , Y may be expressed asP ( X , Y , X x , X y , Y x , Y y , Y xy , Y yy).Proposition 5.9. An infinitesimal Lie symmetry X ∂∂x + Y ∂∂y of (E 1) isuniquely determined by the eight initial Taylor coefficients:(5.10) X (0), Y (0), X x (0), X y (0), Y x (0), Y y (0), Y xy (0), Y yy (0).251

252The bound dim SYM(E 1 ) 8 is attained with F = 0, whence all P = 0and⎧A := ∂ y , E := y ∂ y ,⎪⎨ B := ∂ x , F := y ∂ x ,(5.11)C := x∂ y , G := xx∂ x + xy ∂ y ,⎪⎩D := x∂ x , H := xy ∂ x + yy ∂ y .are infinitesimal generators of the group PGL 3 (K) = Aut(P 2 (K)) of projectivetransformations(5.12) (x, y) ↦→( )αx + βy + γλx + µy + ν , δx + ηy + ǫλx + µy + ν ,stabilizing the collections of all affine lines of K 2 , namely the solutions ofthe model equation y xx = 0. The model Lie algebra pgl 3 (K) ≃ sl 3 (K) issimple.Theorem 5.13. The bound dim SYM(E 1 ) 8 is attained if and only if(E 1 ) is equivalent, through a diffeomorphism (x, y) ↦→ (X, Y ), to Y XX = 0.Proof. The statement is well known ([Lie1883, EL1890, Tr1896, Se1931,Ca1932a, Ol1986, HK1989, Ib1992, Ol1995, Sh1997, Su2001, N2003,Me2004]). We provide a (new?) proof which has the advantage to enjoydirect generalizations to all PDE systems whose model Lie algebras aresemisimple, for instance (E 2 ), (E 3 ) and (E 5 ).The Lie brackets between the eight generators (5.11) are:A B C D E F G HA 0 0 0 0 A B C D + 2EB 0 0 A B 0 0 E + 2D FC 0 −A 0 −C C D − E 0 GD 0 −B C 0 0 −F G 0E −A 0 −C 0 0 F 0 HF −B 0 −D + E F −F 0 H 0G −C −E − 2D 0 −G 0 H 0 0H −D − 2E −F −G 0 −H 0 0 0Table 2.

Assuming that dimSYM(E 1 ) = 8, taking account of (5.7), after makingsome linear combinations, there must exist eight generators of the form⎧A ′ := ∂ y + O(1), E ′ := y ∂ y + O(2),⎪⎨ B ′ := ∂ x + O(1), F ′ := y ∂ x + O(2),(5.14)C ′ := x∂ y + O(2), G ′ := xx∂ x + xy ∂ y + O(3),⎪⎩D ′ := x∂ x + O(2), H ′ := xy ∂ x + yy ∂ y + O(3).To insure that the Lie brackets between these vector fields are small perturbationsof the model ones, we can in advance replace (x, y) by (εx, εy), sothat y xx = ε F ( εx, εy, y x)is an O(ε), hence all the remainders O(1), O(2)and O(3) above are also O(ε). It follows that the structure constants forA ′ , . . .,H ′ are ε-close to those of Table 2.Theorem 5.15. ([OV1994]) Every semisimple Lie algebra over R or C isrigid: small deformations of the structure constants just give isomorphic Liealgebras.Consequently, there exists a change of basis close to the identity leadingto new generators A ′′ , B ′′ , . . .,G ′′ , H ′′ having exactly the same structureconstants as in Table 2. Then A ′′ (0) and B ′′ (0) are still linearly independent.Since [ A ′′ , B ′′] = [A, B] = 0, there exist local coordinates (X, Y ) centeredat 0 in which A ′′ = ∂ X and B ′′ = ∂ Y . Since [ A ′′ , C ′′] = [A, C] = 0and [ B ′′ , C ′′] = [B, C] = A, it follows that C ′′ = X∂ Y . The tangency to0 = −Y 2 + F(X, Y, Y 1 ) (with F(0) = 0) of ( ∂ X) (2)= ∂ X , of ( ∂ Y) (2)= ∂ Yand of ( X∂ Y) (2)= X∂ Y + ∂ Y1 yields F = 0.Open question 5.16. Does this proof generalize to y x κ+1 =F ( x, y, y x , . . ., y x κ)?5.17. Complete system of second order. We now summarize a generalizationto (E 2 ). According to Section 7 below, one may assume that thesubmanifold of solutions is y = b + ∑ ni=1 ai[ x i + O(|x| 2 ) + O(a) + O(b) ] (,whence y x i 1x i 2 = F i1 ,i 2 x i , y,(y x k)with) F(x, y, 0) ≡ 0. Applying to theskeleton 0 = −y i1 ,i 2+ F i1 ,i 2 x i , y, y k a second prolongation L (2) havingcoefficients Y i1 given by (3.9)(II) and Y i1 ,i 2given by (3.20)(II), we get(5.18) 0 = −Y i1 ,i 2+n∑k=1[ ] Xk∂F i1 ,i 2+ [ Y ] ∂F i1 ,i 2∂x k ∂y+n∑k=1253[Yk]∂F i1 ,i 2∂y k.Replacing y i1 ,i 2everywhere by F i1 ,i 2= y 1 R + · · · + y n R, developping inpowers of the pure jet variables y l and picking the coefficients of cst., of y k ,

254of (y k ) 2 and of (y k ) 3 , we get the linear system(5.19)⎧Y x i 1x i 2 = P ( )Y x l⎪⎨ δi k 1Y x i 2y + δi k 2Y x i 1y − X kx i 1x i 2= P ( Y y , X l 2, X )l , Y , Yx l 1 x lδ k,ki 1 ,i 2Y yy − δi k 1X kx⎪⎩i 2y − δk i 2X kx i 1y = P( Y y , X l 2, X )lx l 1 y , X l , Y , Y x lδ k,ki 1 ,i 2Xyy k = P( Y y , X l 2, X lx l 1 y , X l , Y , Y x l),upon which obvious linear combinations yield a known generalization ofProposition 5.9.∑Proposition 5.20. ([Su2001, GM2003a]) An infinitesimal Lie symmetrynk=1 X k ∂ + Y ∂ is uniquely determined by the ∂x k ∂y n2 + 4n + 3 initialTaylor coefficients:(5.21) X l (0), Y (0), X l 2(0), X lx l 1 y (0), Y x l(0), Y y(0), Y x y(0), Y l yy (0).The bound dimSYM(E 2 ) n 2 + 4n + 3 is attained with F i1 ,i 2= 0,whence all P = 0 and(5.22)⎧A := ∂ y , E := y ∂ y ,⎪⎨ B i := ∂ x i, F i := y ∂ x i,C i := x i ∂ y , G i := x i( )x 1 ∂ x 1 + · · · + x n ∂ x n + y ∂ y + xy ∂y ,⎪⎩D i,k := x i ∂ x k, H := y ( )x 1 ∂ x 1 + · · · + x n ∂ x n + y ∂ y .are infinitesimal generators of the group PGL n+2 (K) = Aut(P n+1 (K)) ofprojective transformations(5.23)(x, y) ↦→(α1 x 1 + · · · + α n x n + βy + γλ 1 x 1 + · · · + λ n x n + µy + ν ,)δ 1 x 1 + · · · + δ n x n + ηy + ǫλ 1 x 1 + · · · + λ n x n + µy + ν ,stabilizing the collections of all affine planes of K n+1 , namely the solutionsof the model equation y x i 1x i 2 = 0. The model Lie algebra pgl n+2 (K) ≃sl n+2 (K) is simple, hence rigid.Theorem 5.24. The bound dim SYM(E 2 ) n 2 +4n+3 is attained if andonly if (E 2 ) is equivalent, through a diffeomorphism (x i , y) ↦→ (X k , Y ), toY X k 1X k 2 = 0.The proof, similar to that of Theorem 5.13, is skipped.The study of (E 3 ) also leads to the model algebra pgl n+2 (K) ≃ sl n+2 (K)and an analog to Theorem 5.13 holds. Details are similar.§6. TRANSFER OF LIE SYMMETRIES TO THE PARAMETER SPACE6.1. Stabilization of foliations. As announced in §2.38, we now transferthe theory of Lie symmetries to submanifolds of solutions.

Restarting from §4.1, let ϕ a Lie symmetry of (E ), namely ϕ ∆E stabilizesF ∆E . The diffeomorphism A defined by (2.9) transforms F v to F ∆E . Conjugating,we get the self-transformation A −1 ◦ϕ ∆E ◦A of the (x, a, b)-space thatmust stabilize also the foliation F v . Equivalently, it must have expression:(6.2) [A −1 ◦ ϕ ∆E ◦ A ] (x, a, b) = ( θ(x, a, b), f(a, b), g(a, b) ) ∈ K n × K p × K m ,where, importantly, the last two components are independent of the coordinatex, because the leaves of F v are just {a = cst., b = cst}.Lemma 6.3. To every Lie symmetry ϕ of (E ), there corresponds a transformationof the parameters(6.4) (a, b) ↦−→ ( f(a, b), g(a, b) ) =: h(a, b)meaning that ϕ transforms the local solution y a,b (x) := Π(x, a, b) to thelocal solution y h(a,b) (x) = Π(x, h(a, b)).Unfortunately, the expression of A −1 ◦ϕ ∆E ◦A does not clearly show thatf and g are independent of x. Indeed, reminding the expressions of A andof Φ, we have:(6.5) (ϕ ∆E ◦A(x, a, b) =ϕ(x, Π(x, a, b)), Φ j(q)β(q)255(xi 1, Π j 1(x, a, b), Π j(q 1)x β(q 1 ) (x, a, b) )) .To compose with A −1 whose expression is given by (2.21), it is useful tosplit ϕ = (φ, ψ) ∈ K n × K m , so above we write(6.6) ϕ(x, Π(x, a, b)) = ( φ(x, Π(x, a, b)), ψ(x, Π(x, a, b)) ) ,and finally, droping the arguments:(6.7)[A −1 ◦ ϕ ∆E ◦ A ] (x, a, b) =(φ i , A q( φ i 1, ψ j 1, Φ j(q 1)β(q 1 )), Bj ( φ i 1, ψ j 1, Φ j(q 1)β(q 1 )) ) .In case (E ) = (E 1 ), is an exercise to verify by computations that the A q (·)and B j (·) are independent of x. In general however, the explicit expressionof Φ j i 1 ,...,i λis unknown. Unfortunately also, nothing shows how( )f(a, b), g(a, b) is uniquely associated to ϕ(x, y). Further explanations areneeded.6.8. Determination of parameter transformations. At first, we state ageometric reformulation of the preceding lemma.Lemma 6.9. Every Lie symmetry (x, y) ↦→ ϕ(x, y) of (E ) induces a localK-analytic diffeomorphism(6.10) (x, y, a, b) ↦−→ ( ϕ(x, y), h(a, b) )

256of K n x × K m y × K p a × K m bsolutionsthat maps to itself the associated submanifold of(6.11) M E = { (x, y, a, b) : y = Π(x, a, b) } .Proof. In fact, we know that the n-dimensional leaf {( x, Π(x, a, b) ) : x ∈K n} is sent {( x, Π(x, h(a, b)) ) : x ∈ K n} .Equivalently, setting c := (a, b) and writing (ϕ, h) = (φ, ψ, h), we haveψ = Π(φ, h) when y = Π(x, c), namely(6.12) ψ(x, Π(x, c)) ≡ Π ( φ(x, Π(x, c)), h(c) )Proposition 6.13. There exists a universal rational map Ĥ such that()(6.14) h(c) ≡ Ĥ J κ+1x,a,b Π(x, c), Jκ x,y ϕ(x, Π(x, c))This shows unique determination of h from ϕ, given (E ) or equivalently,given Π.Proof. Differentiating a function χ(x, Π(x, c)) with respect to x k , k =1, . . ., n, corresponds to applying to χ the vector field(6.15) L k := ∂∂x k+m∑j=1∂Π j(x, c) ∂ , k = 1, . . ., n.∂x k ∂yj Thus, applying L k to the m scalar equations (6.12), we get(6.16) L k ψ j =n∑l=1∂Π j∂x l L k φ l ,for 1 k n and 1 j m. It follows from the assumption that ϕ is alocal diffeomorphism that det ( L k φ l (0) ) 1ln≠ 0 also. So we may solve1knthe first derivatives Π x above: there exist universal polynomials S j lsuch that(∂Π j S j {Lk } ) 1i′lϕ i′ ′ n+m1k(6.17)=′ n∂x l det ( L k ′ φ l′) .1l ′ n1k ′ nAgain, we apply the L k to these equations, getting, thanks to the chain rule:( {Lk } )n∑R j 1i′l 1 ,k 1L k ′2ϕ i′ ′ n+m1k 1(6.18)′ ,k′ 2] n 2.l 2 =1∂ 2 Π j∂x l 1 xl 2L k φ l 2=[det ( L k ′ φ l′) 1l ′ n1k ′ n

Here, R j l 1 ,kare universal polynomials.Π j , we getx l 1x l 2(6.19)∂ 2 Π j∂x l 1 xl 2=By induction, for every β ∈ N n :(6.20)∂ |β| Π j∂x β =257Solving the second derivatives( {Lk } )S j 1i′l 1 ,l 2 1L k ′2ϕ i′ ′ n+m1k 1 ′ ,k′ 2[n det ( L k ′ φ l′) ]1l ′ n 3.1k ′ n[S j β( {L } )β ′ 1iϕ i′ ′ n+m|β ′ ||β|det ( L k ′ φ l′) ]1l ′ n 2|β|+1,1k ′ nwhere S j β are universal polynomials. Here, for β′ ∈ N n , we denote by L β′the derivation of order |β ′ | defined by (L 1 ) β′ 1 · · ·(Ln ) β′ n .Next, thanks to the assumption that M is solvable with respect to theparameters, there exist integers j(1), . . ., j(p) with 1 j(q) m and multiindicesβ(1), . . ., β(p) ∈ N n with |β(q)| 1 and max 1qp |β(q)| = κsuch that the local K-analytic map(6.21) ((ΠK p+m ∋ c ↦−→ j (0, c) ) ( ) )1jm ∂ |β(q)| Π j(q), (0, c) ∈ K p+m∂x β(q) 1qphas rank p + m at c = 0. We then consider in (6.20) only the (p + m)equations written for (j, 0), (j(q), β(q)) and we solve h(c) by means of theanalytic implicit function theorem:(6.22)h = Ĥ⎛⎜⎝φ,( {LS j(1) } ) (β ′ 1iβ(1)ϕ i′ ′ n+m{LS j(p) } ) ⎞β ′ 1i|β ′ ||β(1)|β(p)ϕ i′ ′ n+m|β[ (Lkdet ′ φ l′) ]1l ′ n 2|β(1)|+1, . . .,′ ||β(p)| ⎟[ (Lkdet ′ φ l′) ]1l ′ n 2|β(p)|+1⎠ .1k ′ n1k ′ nFinally, by developping every derivative L β′ ϕ i′ (including L k ′φ l′ as a specialcase), taking account of the fact that the coefficients of the L k ′ depend(|βdirectly on Π, we get some universal polynomial P β ′ J ′ |+1x Π, J |β′ |)x,y ϕ i′ .Inserting above, we get the map Ĥ.6.23. Pseudogroup of twin transformations. The previous considerationslead to introducing the following.Definition 6.24. By G v,p , we denote the infinite-dimensional (pseudo)groupof local K-analytic diffeomorphisms(6.25) (x, y, a, b) ↦−→ ( ϕ(x, y), h(a, b) )that respect the separation between the variables and the parameters.

258A converse to Lemma 6.3 holds.Lemma 6.26. Let M be a submanifold y = Π(x, a, b) that is solvablewith respect to the parameters (a, b). If a local K-analytic diffeomorphism(x, y, a, b) ↦−→ ( ϕ(x, y), h(a, b) ) of K n x × K m y × K p a × K m b belonging to G v,psends M to M , then (x, y) ↦→ ϕ(x, y) is a Lie symmetry of the PDE systemE M associated to M .Proof. In fact, since (ϕ, h) respects the separation of variables and stabilizesM , it respects the fundamental pair of foliations ( F v , F p), namely {(a, b) =(a 0 , b 0 )} ∩M is sent to {(a, b) = h(a 0 , b 0 )} ∩ M and {(x, y) = (x 0 , y 0 )} ∩M is sent to {(x, y) = ϕ(x 0 , y 0 )} ∩ M . Hence ϕ ∆EM also stabilizes F ∆E .Corollary 6.27. Through the one-to-one correspondence (E ) ←→ M ofProposition 2.17, Lie symmetries of (E ) correspond to elements of G v,pwhich stabilize M .Definition 6.28. Let Aut v,p (M ) denote the local (pseudo)group of (ϕ, h) ∈G v,p stabilizing M . Let Lie(E ) denote the local (pseudo)group of Lie symmetriesof (E ).In summary:(6.29) Lie(E ) ≃ Aut v,p(M(E))and Aut v,p (M ) ≃ Lie ( E M).6.30. Transfer of infinitesimal Lie symmetries. Let L ∈ SYM(E ), i.e.L ∆E is tangent to ∆ E . Through the diffeomorphism A, the push-forward ofL ∆E must be of the form(6.31)A −1∗ (L ∆ E) =n∑k=1Θ i (x, a, b) ∂∂x i + p∑q=1F q (a, b) ∂∂a q + m∑j=1G j (a, b) ∂∂b j ,where the last two families of K-analytic coefficients F q and G j dependonly on (a, b).Lemma 6.32. To every infinitesimal symmetry L of (E ), we can associatean infinitesimal symmetryp∑(6.33) L ∗ := F q (a, b) ∂∂a + ∑ mG j (a, b) ∂q ∂b jq=1of the space of parameters which tells how the flow of L acts infinitesimallyon the leaves of F ∆E . Furthermore, L + L ∗ is tangent to the submanifoldof solutions M (E ) .Considering the flow of L + L ∗ reduces these assertions and the next tothe arguments of the preceding paragraphs. So we summarize.j=1

Lemma 6.34. Let M be a submanifold y = Π(x, a, b) that is solvable withrespect to the parameters (a, b). If a vector field that respects the separationbetween variables and parameters, namely of the form(6.35)L +L ∗ =n∑i=1X i (x, y) ∂∂x i+ m∑j=1Y j (x, y) ∂∂y j + p∑q=1259F q (a, b) ∂∂a q + m∑is tangent to M , then L is an infinitesimal Lie symmetry of ( E M)Corollary 6.36. Through the one-to-one correspondence (E ) ←→ M ofProposition 2.17, infinitesimal Lie symmetries of (E ) correspond to vectorfields L + L ∗ tangent to M .Definition 6.37. Let SYM(M ) denote the Lie algebra of vector fields L +L ∗ tangent to M . Let SYM(E ) denote the Lie algebra of infinitesimal Liesymmetries of (E ).In summary:(6.38)SYM(E ) ≃ SYM ( M (E ))j=1and SYM ( M ) ≃ SYM ( E M).G j (a, b) ∂∂b j6.39. Dual defining equations. As in §2.10, let M ⊂ K n x ×Km y ×Kp a ×Km bgiven by 0 = −y j + Π j (x, a, b) and assume if to be solvable with respectto the parameters. In particular, we can solve the b j , obtaining dual definingequations(6.40) b j = Π ∗j (a, x, y), j = 1, . . .,m,for some local K-analytic map map Π ∗ = (Π ∗1 , . . .,Π ∗m ) satisfying(6.41) b ≡ Π ∗( a, x, Π(x, a, b) ) and y ≡ Π ( x, a, Π ∗ (a, x, y) ) .6.42. An algorithm for the computation of SYM(M ). The tangency toM is expressed by applying the vector field (6.35) to 0 = −y j +Π j (x, a, b),which yields:(6.43)0 = −Y j (x, y) +n∑X i (x, y) Π j (x, a, b) +x ii=1+p∑q=1F q (a, b) Π j aq(x, a, b)m∑G l (a, b) Π j (x, a, b),b lfor j = 1, . . ., m and for (x, y, a, b) ∈ M . In fact, after replacing the variabley by Π(x, a, b), these equations should be interpreted as power seriesidentities in K{x, a, b}.l=1

260(Denote by ∆(x, a, b) the determinant of the (invertible) matrixΠj(x, a, b) ) and by D(x, a, b) its matrix of cofactors, so thatb l 1l,jmΠ −1b= [∆] −1 D. Hence we can solve G from (6.43):(6.44) ⎧[D(x, a, b)G (a, b) ≡ Y ( x, Π(x, a, b) ) n∑− X ⎪⎨i( x, Π(x, a, b) ) Π x i(x, a, b)−∆(x, a, b)i=1]p∑− F ⎪⎩q (a, b) Π a q(x, a, b) .q=1Next, we aim to solve the ( F q (a, b). Consequently, we gather all the otherterms in the brackets as Ψ 0 J1x,a,b Π, X , Y ) :(6.45)G (a, b) ≡[D(x, a, b)−∆(x, a, b)]p∑F q (a, b) Π a q(x, a, b)q=1+ Ψ (0 J1x,a,b Π, X , Y ).∆(x, a, b)Here, Ψ 0 is linear with respect to (X , Y ), with polynomial coefficients ofdegree one in Jx,a,b 1 Π.Next, for k = 1, . . ., n, we differentiate this identity with respect to x k .Then G (a, b) disappears and we chase the denominator ∆ 2 :(6.46) ⎧ []p∑0 ≡ [∆ D] − F q (a, b) Π a q x k(x, a, b) +⎪⎨q=1[]p∑+ [∆ D xk − ∆ xk D] − F q (a, b) Π a q(x, a, b) +⎪⎩q=1+ Ψ k(J2x,a,b Π, J 1 x,y X , J1 x,y Y ) .The Ψ k are linear with respect to (Jx,y 1 X , J1 x,yY ), with polynomial coefficientsin Jx,a,b 2 Π. Then we further differentiate with respect to x and byinduction, for every β ∈ N n , we get:(6.47) ⎧ []p∑0 ≡ [∆ D] − F q (a, b) Π a q x β(x, a, b) +q=1⎪⎨+ ∑ (D β,β1 J|β 1 |+1 Π )[ ]p∑− F q (a, b) Π a q x 1(x, a, b) +β⎪⎩|β 1 |

where the expressions D β,β1 are certain m × m matrices withpolynomial coefficients in the jet J |β 1|+1x,a,bΠ, and where the terms(|β|+1 |β|Ψ β Jx,a,bΠ, J x,yX , J x,yY |β| ) are linear with respect to ( J x,yX |β| , J x,yY |β| ) ,with polynomial coefficients in J |β|+1x,a,b Π.Writing these identity for (j, β) = (j(q), β(q)), q = 1, . . .,p, remindingmax 1qp |β(q)| = κ, it follows from the assumption of solvability withrespect to the parameters (a boring technical check is needed) that we maysolve(6.48)F q (a, b) ≡ Φ q( J κ+1x,a,b Π(x, a, b), Jκ x,yX (x, Π(x, a, b)), Jx,yY κ (x, Π(x, a, b)) ) ,for q = 1, . . .,p, where each local K-analytic function Φ q is linear with respectto (J κ X , J κ Y ) and rational with respect to J κ+1 Π, with denominatornot vanishing at (x, a, b) := (0, 0, 0).Pursuing, we differentiate (6.48) with respect to x l for l = 1, . . .,n. ThenF q (a, b) disappears and we get:(6.49) (0 ≡ Φ q,l Jκ+2x,a,bΠ(x, a, b), Jκ+1 x,y X (x, Π(x, a, b)), Jκ+1 x,y Y (x, Π(x, a, b))) ,for 1 q p and 1 l n. In (6.46), we then replace the functions F qby their values Φ q :(6.50)0 ≡ Ψ k,j(Jκ+1x,a,b Π(x, a, b), Jκ x,y X (x, Π(x, a, b)), Jκ x,y Y (x, Π(x, a, b))) ,for 1 k n and 1 j m. Then we replace the variable b by Π ∗ (a, x, y)in the two obtained systems (6.49) and (6.50); taking account of the functionalidentity y ≡ Π ( x, a, Π ∗ (a, x, y) ) written in (6.41), we get(6.51) {0 ≡ Φq,l(Jκ+2x,a,b Π(x, a, Π∗ (a, x, y)), Jx,y κ+1X (x, y), Jκ+1 x,y Y (x, y)) ,(0 ≡ Ψ k,j Jκ+1x,a,b Π(x, a, Π∗ (a, x, y)), Jx,y κ X (x, y), Jκ x,y Y (x, y)) .Finally, we develope these equations in power series with respect to a:(6.52)⎧⎪⎨0 ≡ ∑ (a γ Φ q,l,γ x, y, Jκ+1x,y X (x, y), Jκ+1 x,y Y (x, y)) ,γ∈N p0 ≡ ∑ (a ⎪⎩γ Ψ k,j,γ x, y, Jκx,y X (x, y), Jx,y κ Y (x, y)) ,γ∈N pwhere the terms Φ q,l,γ and Ψ k,j,γ are linear with respect to the jets of X , Y .Proposition 6.53. A vector field (6.35) belongs to SYM(M ) if and only ifX i , Y j satisfy the linear PDE system{ (0 ≡ Φq,l,γ x, y, Jκ+1x,y X (x, y), Jκ+1 x,y(6.54)Y (x, y)) ,(0 ≡ Π k,j,γ x, y, Jκx,y X (x, y), Jx,y κ Y (x, y)) ,261

262where 1 q p, 1 l n, 1 k n and γ ∈ N p . Then F q definedby (6.48) and G j defined by (6.45) are independent of x.This provides a second algorithm, essentially equivalent to Sophus Lie’s.Example 6.55. For y xx (x) = F(x, y(x), y x (x)), the first line of (6.54) is (thesecond one is redundant):(6.56) ⎧0 ≡ X [−Π xa Π xxx Π b + Π a Π xb Π xxx − Π x Π xxa Π xb + Π xa Π x Π xxb +⎪⎨⎪⎩+ Y [−Π xa Π xxb + Π xxa Π xb ] ++Π xxa Π xx Π b − Π a Π xxb Π xx ] ++ X x [−2Π xx Π xa Π b + 2Π xx Π a Π xb + Π x Π b Π xxa − Π x Π a Π xxb ] ++ Y x [−Π b Π xxa + Π a Π xxb ] ++ X y[−3Πx Π xx Π xa Π b + 3Π x Π a Π xx Π xb + (Π x ) 2 Π b Π xxa − (Π x ) 2 Π a Π xxb]++ Y y [Π xx Π b Π xa − Π xx Π a Π xb − Π x Π b Π xxa + Π x Π a Π xxb ] ++ X xx [−Π x Π b Π xa + Π x Π a Π xb ] +[ ]+ X xy −2(Πx ) 2 Π b Π xa + 2(Π x ) 2 Π a Π xb +[ ]+ X y 2 −(Πx ) 3 Π b Π xa + (Π x ) 3 Π a Π xb ++ Y xx [Π b Π xa − Π a Π xb ]++ Y xy [2Π x Π b Π xa − 2Π x Π a Π xb ]+[ ]+ Y y 2 (Πx ) 2 Π b Π xa − (Π x ) 2 Π a Π xb .We observe the similarity with (4.19): the expression is linear in the partialderivatives of X , Y of order 2, but the coefficients in the equation aboveare more complicated. In fact, after dividing by −Π b Π xa + Π a Π xb , thisequation coincides with (4.21), thanks to Π x = y 1 and to the formulas (2.34)for F x , F y , F y1 .6.57. Infinitesimal CR automorphisms of generic submanifolds. If thesystem (E ) is associated to the complexification M = (M) c of a genericM ⊂ C n+m as in §1.16, then a = (¯z) c = ζ, b = ( ¯w) c = ξ, and the vectorfield L ∗ associated to an infinitesimal Lie symmetry(6.58) L =n∑i=1X i (z, w) ∂∂z i + m∑i=1j=1Y j (z, w)∂∂w jof (E ) is simply the complexification L of its conjugate L , namelyn∑(6.59) L ∗ = L = X i (ζ, ξ) ∂∂ζ + ∑ mY j (ζ, ξ) ∂i ∂ξ . jj=1

Then the sum L + L is tangent to M and its flow stabilizes the two invariantfoliations, obtained by intersecting M by {(z, w) = cst.} or by{(ζ, ξ) = cst.}. In [Me2005a, Me2005b], these two foliations, denotedF , F , are called (conjugate) Segre foliations, since its leaves are the complexificationsof the (conjugate) classical Segre varieties ([Se1931, Pi1975,Pi1978, We1977, DW1980, BJT1985, DF1988, BER1999, Su2001, Su2002,Su2003, GM2003a]) associated to M, viewed in its ambient space C n+m .The next definition is also classical ([Be1979, Lo1981, EKV1985, Kr1987,KV1987, Be1988, Vi1990, St1996, Be1997, BER1999, Lo2002, FK2005a,FK2005b]):Definition 6.60. By hol(M) is meant the Lie algebra of local holomorphicvector fields L = ∑ ni=1 X i (z, w) ∂ + ∑ m∂z i j=1 Y j (z, w) ∂ whose real∂w jflow exp ( tL ) (z, w) induces one-parameter families of local biholomorphictransformations of C n+m stabilizing M. Equivalently,(6.61) 2 ReL = L + Lis tangent to M. Again equivalently, L + L is tangent to M = M c .Then obviously hol(M) is a real Lie algebra.Theorem 6.62. ([Ca1932a, BER1999, GM2004]) The complexificationhol(M) ⊗ C identifies with SYM ( E (M c ) ) . Furthermore, if M is finitelynondegenerate and minimal at the origin, both are finite-dimensional andhol(M) is totally real in SYM ( E (M c ) ) .The minimality assumption is sometimes presented by saying that the Liealgebra generated by T c M generates TM at the origin ([BER1999]). However,it is more natural to proceed with the fundamental pair of foliationsassociated to M ([Me2001, GM2004, Me2005a, Me2005b]). AnticipatingSections 10 and 11 to which the reader is referred, we set.Definition 6.63. A real analytic generic submanifold M ⊂ C n+m is minimalat one of its points p if the fundamental pair of foliations of its complexificationM is covering at p (Definition 10.17).Further informations may be found in Section 10. We conclude by formulatingapplications of Theorems 5.13 and 5.24.Corollary 6.64. The bound dim hol(M) 8 for a Levi nondegenerate hypersurfaceM ⊂ C 2 is attained if and only if it is locally biholomorphic tothe sphere S 3 ⊂ C 2 .Corollary 6.65. The bound dim hol(M) n 2 + 4n + 3 for a Levi nondegeneratehypersurface M ⊂ C n+1 is attained if and only if it is locallybiholomorphic to the sphere S 2n+1 ⊂ C n+1 .263

264§7. EQUIVALENCE PROBLEMS AND NORMAL FORMS7.1. Equivalences of submanifolds of solutions. As in §3.1, let (E ) and(E ′ ) be two PDE systems and assume that ϕ transforms (E ) to (E ′ ). DefiningA ′ similarly as A, it follows that(7.2) A ′ −1 ◦ Φ E ,E ′ ◦ A(x, a, b) ≡ ( θ(x, a, b), f(a, b), g(a, b) ) =: (x ′ , a ′ , b ′ )transforms F v to F ′ v, hence induces a map (a, b) ↦→ (a ′ , b ′ ). The arguments ofSection 6 apply here with minor modifications to provide two fundamentallemmas.Lemma 7.3. Every equivalence (x, y) ↦→ (x ′ , y ′ ) between to PDE systems(E ) and (E ′ ) comes with an associated transformation (a, b) ↦→ (a ′ , b ′ ) ofthe parameter spaces such that(7.4) (x, y, a, b) ↦−→ (x ′ , y ′ , a ′ , b ′ )is an equivalence between the associated submanifolds of solutions M (E ) →M ′(E ′ ) .Conversely, let M and M ′ be two submanifolds of K n x × K m y × K p a × K m band of K n x ′ × Km y ′ × Kp a ′ × K m b ′ represented by y = Π(x, a, b) and by y′ =Π ′ (x ′ , a ′ , b ′ ), in the same dimensions. Assume both are solvable with respectto the parameters.Lemma 7.5. Every equivalence(7.6) (x, y, a, b) ↦−→ ( ϕ(x, y), h(a, b) )between M and M ′ belonging to G v,p induces by projection the equivalence(x, y) ↦→ ϕ(x, y) between the associated PDE systems ( E M)and(E′M ′).7.7. Classification problems. Consequently, classifying PDE systems underpoint transformations (Section 3) is equivalent to the following.Equivalence problem 7.8. Find an algorithm to decide whether two givensubmanifolds (of solutions) M and M ′ are equivalent through an elementof G v,p .Classification problem 7.9. Classify submanifolds (of solutions) M ,namely provide a complete list of all possible such equations, including theirautomorphism group Aut v,p (M ) ⊂ G v,p .7.10. Partial normal forms. Both problems above are of high complexity.At least as a preliminary step, it is useful to try to simplify somehowthe defining equations of M , by appropriate changes of coordinates belongingto G v,p . To begin with, the next lemma holds for M defined byy = Π(x, a, b) with the only assumption that b ↦→ Π(0, 0, b) has rank m atb = 0.

Lemma 7.11. ([CM1974, BER1999, Me2005a], [∗]) In coordinates x ′ =(x ′1 , . . .,x ′n ) and y ′ = (y ′1 , . . .,y ′m ) an arbitrary submanifold M ′ definedby y ′ = Π ′ (x ′ , a ′ , b ′ ) or dually by b ′ = Π ′∗ (a ′ , x ′ , y ′ ) is equivalent to(7.12) y = Π(x, a, b) or dually to b = Π ∗ (a, x, y)with(7.13)Π(0, a, b) ≡ Π(x, 0, b) ≡ b or dually Π ∗ (0, x, y) ≡ Π ∗ (a, 0, y) ≡ y,namely Π = b + O(xa) and Π ∗ = y + O(ax).Proof. We develope(7.14) y ′ = Π ′ (0, a ′ , b ′ ) + Λ ′ (x ′ ) + O(x ′ a ′ ).Since b ′ ↦→ Π ′ (0, a ′ , b ′ ) has rank m at b ′ = 0, the coordinate change(7.15) b ′′ := Π ′ (0, a ′ , b ′ ), a ′′ := a ′ , x ′′ := x ′ , y ′′ := y ′ ,transforms M ′ to M ′′ defined by(7.16) y ′′ = Π ′′ (x ′′ , a ′′ , b ′′ ) := b ′′ + Λ ′ (x ′′ ) + O(x ′′ a ′′ ).Solving b ′′ by means of the implicit function theorem, we get(7.17) b ′′ = Π ′′∗ (a ′′ , x ′′ , y ′′ ) = y ′′ − Λ ′ (x ′′ ) + O(a ′′ x ′′ ),and it suffices to set y := y ′′ − Λ ′ (x ′′ ), x := x ′′ and a := a ′′ , b := b ′′ .Taking account of solvability with respect to the parameters, finer normalizationsholds.Lemma 7.18. With n = m = κ = 1, every submanifold of solutions y ′ =b ′ + x ′ a ′[ ]1 + O 1 of y′x ′ x = F ′ (x ′ , y ′ , y ′ ′ x ′) is equivalent to(7.19) y xx = b + xa + O(x 2 a 2 ).Proof. Writing y ′ = b ′ +x ′[ a ′ +a ′ Λ ′ (a ′ , b ′ )+O(x ′ a ′ ) ] , where Λ ′ = O 1 , weset a ′′ := a ′ + a ′ Λ ′ (a ′ , b ′ ), b ′′ := b ′ , x ′′ := x ′ , y ′′ := y ′ , whence y ′′ = b ′′ +x ′′[ a ′′ +O(x ′′ a ′′ ) ] . Dually b ′′ = y ′′ −a ′′[ x ′′ +x ′′ x ′′ Λ ′′ (x ′′ , y ′′ )+O(x ′′ x ′′ a ′′ ) ] ,so we set x := x ′′ + x ′′ x ′′ Λ ′′ (x ′′ , y ′′ ), y := y ′′ , a := a ′′ , b := b ′′ .Corollary 7.20. Every second order ordinary differential equation y x ′ ′ x = ′F ′ (x ′ , y ′ , yx ′ ′) is equivalent to(7.21) y xx = (y x ) 2 R(x, y, y x ).265

2667.22. Complete normal forms. The Moser theory of normal forms may betransferred with minor modifications to submanifolds of solutions associatedto (E 1 ) and to (E 2 ).Theorem 7.23. ([CM1974, Ja1990], [∗]) A local K-analytic submanifold ofsolutions associated to (E 1 ):(7.24) y ′ = b ′ + x ′ a ′ + O 3 = ∑ ∑Π ′ k ′ ,l ′(b′ ) x ′ k ′ a ′ l ′k ′ 0 l ′ 0can be mapped, by a transformation (x ′ , y ′ , a ′ , b ′ ) ↦→ (x, y, a, b) belongingto G v,p , to a submanifold of solutions of the specific form(7.25)y = b + xa + Π 2,4 (b) x 2 a 4 + Π 4,2 (b) a 2 x 4 + ∑ ∑ ∑Π k,l (b) x k a l .k2 l2 k+l7Solving (a, b) from y = Π and y x = Π x with Π as above, we deduce thefollowing.Corollary 7.26. Every y x ′ ′ x = F ′ (x ′ , y ′ , y ′ ′ x ′) is equivalent to(7.27)y xx = (y x ) [ 2 x 2 F 2,2 (y) + x 3 r(x, y) ] + (y x ) [ 4 F 0,4 (y) + x r(x, y) ] ++ ∑ ∑ ∑F k,l (y) x k (y x ) l .k0 l0 k+l5For the completely integrable system (E 2 ) having several dependent variables(x 1 , . . .,x n ), n 2, we have the following.Theorem 7.28. ([CM1974], [∗]) A local K-analytic submanifold of solutionsassociated to (E 2 ):(7.29) y ′ = b ′ + ∑x ′k a ′k + O 31kncan be mapped, by a transformation (x ′ , y ′ , a ′ , b ′ ) ↦→ (x, y, a, b) belongingto G v,p , to a submanifold of solutions of the specific form:(7.30) y = b + ∑x k a k + ∑ ∑Π k,l (x, a, b)1kn k2 l2where(7.31)Π k,l (x, a, b) :=∑∑k 1 +···+k n=k l 1 +···+l n=lwith the terms Π 2,2 , Π 2,3 and Π 3,3 satisfying:Π k1 ,...,k n,l 1 ,...,l n(b) (x 1 ) k1 · · ·(x n ) kn (a 1 ) l1 · · ·(a n ) ln(7.32) 0 = ∆ Π 2,2 = ∆∆ Π 2,3 = ∆∆ Π 3,2 = ∆∆∆ Π 3,3 ,

where(7.33) ∆ := ∑1kn∂ 2∂x k ∂a k .Exercise: solving (a k , b) from y = Π and y x l = Π x l, with Π as above,deduce a complete normal form for (E 2 ).Open problem 7.34. Find complete normal forms for submanifolds of solutionsassociated to (E 4 ) and to (E 5 ).§8. STUDY OF TWO SPECIFIC EXAMPLES8.1. Study of the Lie symmetries of (E 4 ). Its submanifold of solutions possessestwo equations:(8.2) y 1 = Π 1 (x, a, b 1 , b 2 ) y 2 = Π 2 (x, a, b 1 , b 2 ).For instance, a generic submanifold M ⊂ C 3 of CR dimension 1 and ofcodimension 3 has equations of such a form.Assuming V S (E 4 ) to be twin solvable and having covering submanifoldof solutions (see Definition 10.17), it may be verified (for M ⊂ C 3 , see[Be1997]) that at a Zariski-generic point, its equations are of the form:(8.3)y 1 = b 1 + xa + O(x 2 ) + O(b 1 ) + O(b 2 ),y 2 = b 2 + xa(x + a) + O(x 3 ) + O(b 1 ) + O(b 2 ).The model has zero remainders with associated system(8.4) y 2 1 = 2xy1 1 + (y1 1 )2 , y 1 2 = 0,the third equation y2 2 = 2 y1 1 being obtained by differentiating the first.We may put the submanifold in partial normal form. Proceeding asin [BES2005], some partial normalizations belonging to G v,p yield:(8.5)y 1 = b 1 + ax + a 2[ Π 1 3,2(b) x 3 + Π 1 4,2(b) x 4 + · · ·] + O(a 3 x 2 ),y 2 = b 2 + a [ x 2 + Π 2 4,1(b) x 4 + · · ·] + a 2[ x + Π 2 3,2(b) x 3 + · · ·] + O(a 3 x 2 ).Redifferentiating, we get an appropriate, partially normalized system (E 4 ):(8.6) ⎧⎪⎨y1 2 = y1( ) 1 2x + g1+ (y1) 1 2( 1 + g 2) + (y1) 1 3 s + (y1) 1 4 s + (y1) 1 5 s + (y1) 1 6 R,y2 1 ⎪⎩= (y1 1 )2 h + (y1 1 )3 R,y2 2 = )1( y1 2 + g1x + (y11 ) 2( gx 2 + (2x + g1 )h ) + (y1 1 )3 r + (y1 1 )4 r + (y1 1 )5 r + (y1 1 )6 R,where, precisely:• g 1 , g 2 and h are functions of (x, y 1 , y 2 ) satisfying g j = O(xx) +O(y 1 ) + O(y 2 ), j = 1, 2 and h = O(x) + O(y 1 ) + O(y 2 );267

268• r and s are unspecified functions, varying in the context, of (x, y 1 , y 2 )with s = O(x) + O(y 1 ) + O(y 2 ), but possibly r(0) ≠ 0;• R is a remainder function of all the variables (x, y 1 , y 2 , y 1 1) parametrizing∆ E4 .Letting L = X ∂ + Y 1 ∂ + Y 2 ∂ be a candidate infinitesimal Lie∂x ∂y 1 ∂y 2symmetry and applying L (2) = L + Y1 1 ∂ + Y 2 ∂y11 1 ∂ + Y 1 ∂y12 2 ∂ + Y 2 ∂y21 2 ∂∂y22to ∆ E4 , we obtain firstly, computing mod (y1) 1 5 :(8.7)0 ≡ −Y1 2 + [ X ]( y1(2 1 + gx) 1 + (y1) 1 2 gx 2 + (y1) 1 3 r + (y1) 1 4 r ) ++ [ Y 1]( y1 1 r + (y1) 1 2 r + (y1) 1 3 r + (y1) 1 4 r ) ++ [ Y 2]( y1 1 r + (y1) 1 2 r + (y1) 1 3 r + (y1) 1 4 r ) ++ Y1( 1 2x + g 1 + y1 1 (2 + 2g2 ) + (y1 1 )2 s + (y1 1 )3 s + (y1 1 )4 s ) ,and secondly, computing mod (y 1 1 )2 :(8.8) 0 ≡ −Y 1 2 + 2 y1 1 Y1 1 h.The third Lie equation involving Y2 2 will be superfluous. Specializing(4.6)(II) to m = 2, we get Y1 1 and Y1:2(8.9)Y1 1 = Y x 1 + [ Y 1y − X ] 1 x y11 + [ ]Y 1y y2 2 1 + [ − X y 1](y11 ) 2 + [ − X y 2]y11 y12Y1 2 = Y x 2 + [ ]Y 2y y1 1 1 + [ Y 2y − X x]y2 2 1 + [ − X y 1]y21 y1 1 + [ − X y 2](y21 ) 2 .and also Y2 1 and Y2 2 (in fact superfluous):(8.10)Y2 1 = Y xx 1 + [ 2 Y 1xy − X ] 1 xx y11 + [ ]2 Y 1xy y2 2 1 + [ Y 1y 1 y − 2 X ] 1 xy 1 (y11 ) 2 ++ [ 2 Y 1y 1 y − 2 X ] 2 xy 2 y11 y1 2 + [ ]Y 1y 2 y (y2 2 1 ) 2 + [ ]− X y 1 y 1 (y11 ) 3 ++ [ ]− 2 X y 1 y 2 (y11 ) 2 y1 2 + [ ]− X y 2 y 2 y11 (y1 2 )2 + [ Y 1y − 2 X x]y1 1 2 ++ [ ]Y 1y y2 2 2 + [ − 3 X y 1]y11 y2 1 + [ − X y 2]y11 y2 2 + [ − 2 X y 2]y21 y2 1 ,Y2 2 = Yxx 2 + [ ]2 Y 2xy y1 1 1 + [ 2 Y 2xy − X ] 2 xx y21 + [ ]Y 2y 1 y (y1 1 1 ) 2 ++ [ 2 Y 2y 1 y − 2 X ] 2 xy 1 y11 y1 2 + [ Y 2y 2 y − 2 X ] 2 xy 2 (y21 ) 2 + [ ]− X y 1 y 1 (y11 ) 2 y1+2+ [ ]− 2 X y 1 y 2 y11 (y1) 2 2 + [ ]− X y 2 y 2 (y21 ) 3 + [ ]Yy 2 y1 1 2 ++ [ Y 2y − 2 X x]y2 2 2 + [ − 2 X y 1]y11 y2 2 + [ − X y 1]y21 y2 1 + [ − 3 X y 2]y21 y2.2Inserting Y1 2 and Y1 1 in the first Lie equation (8.7) in which y2 1 is replaced bythe value (8.6) 1 it has on ∆ E4 and still computing mod(y1 1)5 , we get, again

with r, s being unspecified functions of (x, y 1 , y 2 ) with s(0) = 0:(8.11)0 ≡ − Yx 2 + [ ]− Y 2y y1 1 1 ++ [ − Y 2y + X x](y1 2 1 (2x + g 1 ) + (y1) 1 2 (1 + g 2 ) + (y1) 1 3 s + (y1) 1 4 s ) ++ [ X y 1]((y11 ) 2 (2x + g 1 ) + (y1 1 )3 (1 + g 2 ) + (y1 1 )4 s ) +269+ [ X y 2]((y11 ) 2 [2x + g 1 ] 2 + (y 1 1 )3 (4x + 2g 1 )(1 + g 2 ) + (y 1 1 )4 (1 + s) ) ++ [ X ]( y 1 1 (2 + g1 x ) + (y1 1 )2 g 2 x + (y1 1 )3 r + (y 1 1 )4 r ) ++ [ Y 1]( y 1 1 r + (y1 1 )2 r + (y 1 1 )3 r + (y 1 1 )4 r ) ++ [ Y 2]( y1 1 r + (y1 1 )2 r + (y1 1 )3 r + (y1 1 )4 r ) ++ [ ](Yx1 2x + g 1 + y1 1 (2 + 2g2 ) + (y1 1 )2 s + (y1 1 )3 s + (y1 1 )4 s ) ++ [ Y 1y − X (x]y1 1 1 (2x + g 1 ) + (y1 1 )2 (2 + 2g 2 ) + (y1 1 )3 s + (y1 1 )4 s ) ++ [ ](Y 1y y1 2 1 [2x + g 1 ] 2 + (y1 1 )2 (2x + g 1 )(3 + 3g 2 ) + (y1 1 )3 (2 + s) + (y1 1 )4 s ) ++ [ − X y 1]((y11 ) 2 (2x + g 1 ) + (y1 1 )3 (2 + 2g 2 ) + (y1 1 )4 s ) ++ [ − X y 2]((y11 ) 2 [2x + g 1 ] 2 + (y 1 1 )3 (2x + g 1 )(3 + 3g 2 ) + (y 1 1 )4 (2 + s) ) .Collecting the coefficients of the monomials cst., y1, 1 (y1) 1 2 , (y1) 1 3 , (y1) 1 4 , weget, after slight simplification (in the coefficient of (y1) 1 2 , the term (2x +g 1 )X x annihilates with its opposite; in the coefficient of (y1 1)3 , two pairsannihilate and then, we divide by [1 + g 2 ]) a system of five linear PDE’s:(8.12)0 = −Yx 2 + (2x + g 1 )Yx 1 ,0 = −Y 2y − (2x + 1 g1 )Y 2y + (2 + 2 g1 x )X + r Y 1 + r Y 2 ++ (2 + 2g 2 )Yx 1 + (2x + g1 )Y 1y + [2x + 1 g1 ] 2 Y 1y 2,0 = −Y 2y + X 2 x + gx[1 2 + g 2 ] −1 X + r Y 1 + r Y 2 ++ s Yx 1 + 2 Y 1y − 2 X 1 x + (6x + 3g 2 )Y 1y 2,0 = s Y 2y + s X 2 x + (1 + g 2 )X y 1 + (2x + g 1 )(2 + 2g 2 )X y 2++ r X + r Y 1 + r Y 2 + s Yx 1 + s Y 1y + s X 1 x + (2 + s) Y 1y 2−− (2 + 2g 2 )X y 1 − (2x + g 1 )(3 + 3g 2 )X y 2,0 = s Y 2y 2 + s X x + s X y 1 + (1 + s)X y 2 + r X + r Y 1 + r Y 2 + s Y 1x + s Y 1y 1++ s X x + s Y 1y 2 + s X y 1 − (2 + s)X y 2.We then simplify the remainders using s + s = s, r + s = r and r + r = r; wedivide (8.12) 5 by (1 + s); we replace X y 2 obtained from (8.12) 5 in (8.12) 4 ;we divide (8.12) 4 by (1 + g 2 ); we then solve X y 1 from (8.12) 4 and finally

270we insert it in (8.12) 5 ; we get:(8.13)0 = −Yx 2 + (2x + g1 )Yx 1 ,0 = −Y 2y − (2x + 1 g1 )Y 2y + (2 + 2 g1 x )X + (2 + 2g2 )Yx 1 + (2x + g1 )Y 1y 1++ [2x + g 1 ] 2 Yy 1 + r Y 1 + r Y 2 ,20 = −Y 2y − X 2 x + 2 Y 1y + (6x + 1 3g2 )Y 1y + r Y 1 + r Y 2 + s Y 12 x ++ g 2 x [1 + g2 ] −1 X ,0 = −X y 1 + (2 + s)Y 1y + r X + r Y 1 + r Y 2 + s X 2 x + s Yx 1 + s Y 1y + s Y 2 1 y 2,0 = −X y 2 + r X + r Y 1 + r Y 2 + s X x + s Yx 1 + s Y 1y + s Y 1 1 y + s Y 2 2 y 2.Similarly, developing the second equation (8.8) and computing mod (y 1 1 )2 ,we get:(8.14)0 ≡ −Y 1xx + [ − 2 Y 1xy + X 1 xx+ [ ]2h Yx1 y11 .]y11 + [ − (4x + 2g 1 )Y xy 2 − (2 + h)Y 1y 2 ]+Collecting the coefficients of the monomials cst., y1 1 , we get two more linearPDE’s:(8.15)0 = −Yxx,10 = −2 Y 1xy + X 1 xx − (4x + 2g 1 )Y 11xy2 − (2 + h)Yy + 2h Y 12Proposition 8.16. Setting as initial conditions the five specific differentialcoefficients(8.17) P := P ( X , Y 1 , Y 2 , Yx 1 , X )x = r X +r Y 1 +r Y 2 +r Yx 1 +r X x,it follows by cross differentiations and by linear substitutions from the sevenequations (8.13) i , i = 1, 2, 3, 4, 5, (8.15) j , j = 1, 2, that X y 1, X y 2, Y 1y, 1Y 1y, Y 2 2 x , Y 2y, Y 2 1 yand X 2 xx , X xy 1, X xy 2, Yxx, 1 Y 1xy, Y 1 1 xyare uniquely2determined as linear combinations of (X , Y 1 , Y 2 , Yx 1 , X x ), namely:⎧Yx⎪⎨2 1= P, X 2 xx = P, Yxx1 3= P,(8.18)⎪⎩X y 1X y 24= P, Y 1y 1 5 = P, Y 2y 1 6 = P, X xy 19= P, Y 1y 2 10= P, Y 2y 2 11= P, X xy 2Then the expressions P are stable under differentiation:(8.19)x .7= P, Y 1xy 1 8 = P,12= P, Y 1xy 2 13= P.P x = P + r Yx 2 + r Yxx 1 + r X xx = P,P y 1 = P + r X y 1 + r Y 1y + r Y 2 1 y + r Y 1 1 xy + r X 1 xy 1 = P,P y 2 = P + r X y 2 + r Y 1y + r Y 2 2 y + r Y 1 2 xy + r X 2 xy 2 = P,

and moreover, all other, higher order partial derivatives of X , of Y 1 and ofY 2 may be expressed as P ( X , Y 1 , Y 2 , Y 1x , X x).Corollary 8.20. An infinitesimal Lie symmetry of (E 4 ) is uniquely determinedby the five initial Taylor coefficients271(8.21) X (0), Y 1 (0), Y 2 (0), Y 1x (0), X x(0).Proof of the proposition. We notice that (8.18) 1 and (8.18) 3 are given forfree by (8.13) 1 and by (8.15) 1 . Differentiating (8.13) 3 with respect to x, weget:(8.22)0 = −Y xy 2 − X xx + 2 Y 1xy + (6 + 1 3g1 x)Y 1y + (6x + 2 3g1 )Y 1xy + r Y 1 +2+ r Yx 1 + r Y 2 + r Yx 2 + r Y x 1 + s Y xx 1 + r X + g2 x [1 + g2 ] −1 X x .By (8.15) 1 , s Yxx 1 vanishes. We replace Yx 2 thanks to (8.13) 1 . Differentiating(8.13) 1 with respect to y 2 , we may substract 0 = −Y xy 2 + (2x + g 1 )Y 1xy+ 2r Yx 1 . We get:(8.23)0 = −X xx + 2 Y 1xy + (4x + 1 2g1 )Y 1xy 2++ (6 + 3gx)Y 1 1y + r X + r Y 1 + r Y 2 + r Y 12 x + gx[1 2 + g 2 ] −1 X x .By means of (8.15) 2 , we replace the first three terms and then solve Y 1y 2 :(8.24) Y 1y 2 = r X + r Y 1 + r Y 2 + r Y 1x + k ∗ X x ,introducing a notation for a new function that should be recorded:(8.25) k ∗ := g 2 x [1 + g2 ] −1 [4 + 3g 1 x − h]−1 .This is (8.18) 10 . Next, we differentiate the obtained equation with respect tox, getting:(8.26) Y 1xy 2 = r X + r Y 1 + r Y 2 + r Y 1x + r X x + k ∗ X xx .This is (8.18) 13 . We replace the obtained value of Y 1yin (8.13) 2 2 , (8.13) 3 ,(8.15) 2 and the obtained value of Y 1xyin (8.15) 2 2 . This yields a new, simpler

272system of seven equations:(8.27)0 = −Yx 2 + (2x + g1 )Yx 1 ,0 = −Y 2y − (2x + 1 g1 )Y 2y + (2 + 2 g1 x )X + (2 + 2g2 )Yx 1 + (2x + g1 )Y 1y 1++ s X + r Y 1 + r Y 2 + s Yx 1 + k ∗ [2x + g 1 ] 2 X x ,0 = −Y 2y − X 2 x + 2 Y 1y + s X + r Y 1 + r Y 2 + s Y 11 x + k∗ (6x + 3g 1 )X x ,0 = −X y 1 + r X + r Y 1 + r Y 2 + s Yx 1 + s X x + s Y 1y + s Y 2 1 y 2,0 = −X y 2 + r X + r Y 1 + r Y 2 + s Yx 1 + s X x + s Y 1y + s Y 2 1 y 2,0 = −Yxx 1 ,0 = −2 Y 1xy + X ( 1 xx 1 − k ∗ (4x + 2g 1 ) ) + r X + r Y 1 + r Y 2 + r Yx 1 . + s X x.Restarting from this system, we differentiate (8.27) 3 with respect to x:(8.28)0 = −Y 2xy − X 2 xx + 2 Y 1xy + r X + r Y 1 + r Y 2 +1+ r X x + r Yx 1 + r Yx 2 + s Yxx 1 + k ∗ (6x + 3g 1 )X xx .We replace Yx 21, we erase Yxx and we add (8.27) 7:(8.29) 0 = −Y 2xy 2 +k∗ (2x+g 1 )X xx +r X +r Y 1 +r Y 2 +r Yx 1 +r X x.We differentiate (8.27) 2 with respect to x:(8.30)0 = −Y 2xy − (2 + 1 g1 x )Y 2y − (2x + 2 g1 )Y 2xy + r X + (2 + 2 g1 x )X x++ s Yx 1 + (2 + 2g2 )Yxx 1 + (2 + g1 x )Y 1y + (2x + 1 g1 )Y 1xy + r X x+1+ s X x + r Y 1 + r Yx 1 + r Y 2 + r Yx 2 + r Yx 1 + s Yxx 1 + k ∗ [2x + g 1 ] 2 X xx .Differentiating (8.27) 1 with respect to y 1 , we may substract 0 = −Y 2xy+ 1(2x + g 1 )Y 1xy+ r Y 1 1 x ; we replace Yx 2 and erase Yxx; 1 we substract (8.29)multiplied by (2x + g 1 ); we get:(8.31) 0 = −Y 2y 2 + (1 + s)X x + Y 1y 1 + r X + r Y 1 + r Y 2 + r Y 1x .Comparing with (8.27) 3 yields:Y 1y(8.32)= (2 + s)X 1 x + r X + r Y 1 + r Y 2 + r Yx 1 ,Y 2y = (3 + s)X 2 x + r X + r Y 1 + r Y 2 + r Yx 1 .These are (8.18) 5 and (8.18) 11 . Differentiating these two equations withrespect to x, replacing Yx 21and erasing Yxx , we get:Y 1xy(8.33)= (2 + s)X 1 xx + r X + r Y 1 + r Y 2 + r Yx 1 + r X x,Y 2xy = (3 + s)X 2 xx + r X + r Y 1 + r Y 2 + r Yx 1 + r X x .We then replace this value of Y 2xy 2 in (8.29) and solve X xx : this yields(8.18) 2 .

To conclude, we replace X xx so obtained in (8.27) 7 : this yields (8.18) 8 .We replace Y 1yand Y 2 1 yfrom (8.32) in (8.27) 2 4 and in (8.27) 5 : this yields(8.18) 4 and this yields (8.18) 9 . Thanks to (8.18) 2 (got) we observe that(8.34) P x = P + r Y 1xx + r Y 2x + r X xx = P.Differentiating (8.18) 4 (got) and (8.18) 9 (got) with respect to x then yields(8.18) 7 and (8.18) 12 . We replace Y 1yand Y 2 1 yfrom (8.18) 2 5 (got) and (8.18) 11(got) in (8.27) 2 : this yields (8.18) 6 . Finally, to obtain the very last (8.18) 13 ,we differentiate (8.18) 10 (got) with respect to x.The proof of Proposition 8.16 is complete.We claim that the bound dimSYM(E 4 ) 5 is attained for themodel (8.4). Indeed, with 0 = r = s and 0 = g 1 = g 2 = h (whencek ∗ = 0) (8.24) is Y 1y= 0 and then the seven equations (8.27) are:2⎧0 = −Yx 2 + 2x Yx 1 ,0 = −Y 2y − 2x Y 21 y + 2 X + 2 Y 2 x 1 + 2x Y 1y 1,⎪⎨ 0 = −Y 2y − X 2 x + 2 Y 1y 1,(8.35) 0 = −X y 1,0 = −X y 2,⎪⎩0 = −Yxx,10 = −2 Y 1xy + X xx, 1having the general solution⎧X = a − d + e x,⎪⎨(8.36)Y 1 = b + d x + 2e y 1 ,⎪⎩Y 2 = c + 2a y 1 + 3e y 2 + d xx.depending on five parameters a, b, c, d, e ∈ K. Five generators of SYM(E 4 )are:⎧D := x∂ x + 2y 1 ∂ y 1 + 3y 2 ∂ y 2,(8.37)The commutator table⎪⎨L 1 := −∂ x + x∂ y 1 + xx∂ y 2,L 1 ′ := ∂ x + 2y 1 ∂ y 2,L 2 := ∂ y 1,⎪⎩L 3 := ∂ y 2.273

274D L 1 L 1 ′ L 2 L 3D 0 −L 1 −L 1 ′ −2 L 2 −3 L 3L 1 L 1 0 −L 2 0 0L 1 ′ L 1 ′ L 2 0 −2 L 3 0L 2 2 L 2 0 2 L 3 0 0L 3 3 L 3 0 0 0 0Table 3.shows that the subalgebra spanned by L 1 , L ′1 , L 2, L 3 is isomorphic tothe unique irreducible 4-dimensional nilpotent Lie algebra n 1 4 ([OV1994,BES2005]). Then SYM(E 4 ) is a semidirect product of K with n 1 4 . Theauthor ignores whether it is rigid. The following accessible research will bepursued in a subsequent publication.Open problem 8.38. Classify systems (E 4 ) up to point transformations. Deducea complete classification, up to local biholomorphisms, of all real analyticgeneric submanifolds of codimension 2 in C 3 , valid at a Zariski-genericpoint.8.39. Almost everywhere rigid hypersurfaces. When studying and classifyingdifferential objects, it is essentially no restriction to assume their Liesymmetry groups to be of dimension 1, the study of objects having no infinitesimalsymmetries being an independent field of research. In particular,if M ⊂ C n+1 (n 1) is a connected real analytic hypersurface, we maysuppose that dimhol(M) 1, at least. So let L be a nonzero holomorphicvector field with L + L tangent to M.Lemma 8.40. ([Ca1932a, St1996, BER1999]) If in addition M is finitelynondegenerate, then(8.41) Σ := { p ∈ M : L (p) ∈ T c pM }is a proper real analytic subset of M.In other words, at every point p belonging to the Zariski-dense subsetM\Σ, the real nonzero vector L (p) + L (p) ∈ T p M supplements Tp cM.Straightening L in a neighborhood of p, there exist local coordinates t =(z 1 , . . .,z n , w) with T0 cM = {w = 0}, T 0M = {Im w = 0}, whence Mis given by Im w = h(z, ¯z, Re w), and with L = ∂ . The tangency of∂w∂+ ∂ = ∂ to M entails that h is indendepent of u. Then the complex∂w ∂ ¯w ∂uequation of M is of the precise form(8.42) w = ¯w + i Θ(z, ¯z),with Θ = 2h simply. The reality of h reads Θ(z, ¯z) ≡ Θ(¯z, z).

Definition 8.43. A real analytic hypersurface M ⊂ C n+1 is called rigid atone of its points p if there exists L ∈ hol(M) with(8.44) T p M = T c pM ⊕ R ( L (p) + L (p) ) .Similar elementary facts hold for general submanifolds of solutions.Lemma 8.45. With n 1 and m = 1, let M be a (connected) submanifoldof solutions that is solvable with respect to the parameters. If there exists anonzero L + L ∗ ∈ SYM(M ), then at Zariski-generic points p ∈ M , wehave L (p) ∉ F v (p) and there exist local coordinates centered at p in whichL = ∂ , L ∗ = ∂ , whence M has equation of the form∂y ∂b(8.46) y = b + Π(x, a),with Π independent of b.The associated system (E M ) has then equations F α that are all independentof y.8.47. Study of the Lie symmetries of (E 5 ). In Example 1.28, it is thusessentially no restriction to assume the hypersurface M ⊂ C 3 to be rigid.Theorem 8.48. ([GM2003b, FK2005a, FK2005b]) The model hypersurfaceM 0 of equation(8.49) w = ¯w + i 2 z1¯z 1 + z 1 z 1¯z 2 + ¯z 1¯z 1 z 21 − z 2¯z 2has transitive Lie symmetry algebra hol(M 0 ) isomorphic to so(3, 2) andis locally biholomorphic to a neighborhood of every geometrically smoothpoint of the tube(8.50) (Rew ′ ) 2 = (Rez ′ 1) 2 + (Rez ′ 1) 3over the standard cone of R 3 . Both are Levi-degenerate with Levi form ofrank 1 at every point and are 2-nondegenerate. The associated PDE system(E M0 )(8.51) y x 2 = 1 4 (y x 1)2 , y x 1 x 1 x 1 = 0(plus other equations obtained by cross differentiation) has infinitesimal Liesymmetry algebra isomorphic to so(5, C), the complexification so(3, 2)⊗C.Through tentative issues ([Eb2006, GM2006]), it has been suspected thatM 0 is the right model in the category of real analytic hypersurfaces M ⊂ C 3having Levi form of rank 1 that are 2-nondegenerate everywhere. Basedon the rigidity of the simple Lie algebra so(5, C) (Theorem 5.15), Theorem8.105 below will confirm this expectation.275

2768.52. Preparation. Thus, translating the considerations to the PDE language,with n = 2 and m = 1, consider a submanifold of solutions ofthe form(8.53)y = b + Π(x, a)= b + 2 x1 a 1 + x 1 x 1 a 2 + a 1 a 1 x 21 − x 2 a 2 + O 4 ,where O 4 is a function of (x, a) only. The term 2 x 1 a 1 corresponds to a Leviform of rank 1 at every point. The term x 1 x 1 a 2 guarantees solvabilitywith respect to the parameters (compare Definition 2.12). Let us develope(8.54) Π(x, a) = ∑ ∑Π k1 ,k 2 ,l 1 ,l 2(x 1 ) k 1(x 2 ) k 2(a 1 ) l 1(a 2 ) l 2,k 1 ,k 2 0 l 1 ,l 2 0with Π k1 ,k 2 ,l 1 ,l 2∈ K. Of course, Π 1,0,1,0 = 2, Π 2,0,0,1 = 1 and Π 0,1,2,0 = 1.Lemma 8.55. A transformation belonging to G v,p insures(8.56)Π k1 ,k 2 ,0,0 = 0, k 1 + k 2 0, Π 0,0,l1 ,l 2= 0, l 1 + l 2 0,Π k1 ,k 2 ,1,0 = 0, k 1 + k 2 2, Π 1,0,l1 ,l 2= 0, l 1 + l 2 2,Π k1 ,k 2 ,2,0 = 0, k 1 + k 2 2, Π 2,0,l1 ,l 2= 0, l 1 + l 2 2.Proof. Lemma 7.11 achieves the first line. The monomial x 1 being factoredby [a 1 +O 2 (a)], we set a 1 := a 1 +O 2 (a) to achieve Π 1,0,l1 ,l 2= 0, l 1 +l 2 2.As in the proof of Lemma 7.18, we pass to the dual equation b = y −Π(x, a)to complete Π k1 ,k 2 ,1,0 = 0, k 1 + k 2 2. Finally, x 1 x 1 is factored by [a 2 +O 2 (a)], so we proceed similarly to achieve the third line.Since Π(x, a) is assumed to be independent of b, the assumption that theLevi form of M ⊂ C 3 has exactly rank 1 at every point translates to:(8.57) 0 ≡∣ Π x 1 a 1 Π ∣x 1 a ∣∣∣ 2Π x 2 a 1 Π .x 2 a 2For later use, it is convenient to develope somehow Π with respect to thepowers of (a 1 , a 2 ):(8.58)y = b + 2 x1 a 1 + x 1 x 1 a 2 + a 1 a 1 x 21 − x 2 a 2 ++ a 2 b(x) + a 1 a 2 d(x) + a 2 a 2 e(x) + a 1 a 1 a 1 f(x) + a 1 a 1 a 2 g(x)++ (a 1 ) 4 R + (a 1 ) 3 a 2 R + a 1 (a 2 ) 2 R + (a 2 ) 3 R,with R = R(x, a) being an unspecified remainder. Thanks to the previouslemma, the coefficients a of a 1 and c of a 1 a 1 must vanish. The function b isan O 3 .

Lemma 8.59. The function b depends only on x 1 , is an O 3 (x 1 ) and thefunction g satisfies g x 2 x2(0) = 0.Proof. Developing [1 −x 2 a 2 ] −1 = 1+x 2 a 2 +(x 2 a 2 ) 2 +O 3 (x 2 a 2 ), insertingthe right hand side of(8.60)y − b = a 1[ 2x 1] + a 2[ x 1 x 1 + b(x) ] + a 1 a 1[ x 2] + a 1 a 2[ 2x 1 x 2 + d(x) ] +277+ a 2 a 2[ x 1 x 1 x 2 + e(x) ] + a 1 a 1 a 1[ f(x) ] + a 1 a 1 a 2[ x 2 x 2 + g(x) ] ++ (a 1 ) 4 R + (a 1 ) 3 a 2 R + a 1 (a 2 ) 2 R + (a 2 ) 3 Rin the determinant (8.57) and selecting the coefficients of cst., of a 1 , of a 2and of a 1 a 1 , we get four PDEs:(8.61)0 = 2b x 2,0 = 2d x 2 − 2b x 1,0 = 4e x 2 − 2x 1 d x 2 − 2x 1 b x 1 − d x 2 b x 1,0 = 2g x 2 − 2d x 1 − [ 6x 1 + 3b x 1]fx 2.The first one yields b = b(x 1 ), which must be an O 3 (x 1 ), because thewhole remainder is an O 4 . Differentiating the fourth with respect to x 2 , itthen follows that g x 2 x2(0) = 0.8.62. Associated PDE system (E 5 ). Next, differentiating (8.60) with respectto x 1 , to x 1 x 1 and to x 1 x 1 x 1 , we compute y x 1 and y x 1 x 1, we substitute y 1 andy 1,1 and we push the monomials a 2 a 2 , a 1 a 1 a 1 and a 1 a 1 a 2 in the remainder:(8.63)y 1 = 2a 1 + a 2 [2x 1 + b x 1] + a 1 a 2 [2x 2 + d x 1] + (a 2 ) 2 R + (a 1 ) 3 R + (a 1 ) 2 a 2 R,y 1,1 = a 2 [2 + b x 1 x 1] + a1 a 2 [d x 1 x 1] + (a2 ) 2 R + (a 1 ) 3 R + (a 1 ) 2 a 2 R,y 1,1,1 = a 2 [b x 1 x 1 x 1] + a1 a 2 [d x 1 x 1 x 1] + (a2 ) 2 R + (a 1 ) 3 R + (a 1 ) 2 a 2 R.Here, the written remainder cannot incorporate a 1 a 1 , so it is said that thecoefficient of a 1 a 1 does vanish in each equation above. Solving for a 1 anda 2 from the first two equations, we get(8.64)⎧a 1 = 1 [ 2x 1 ] [⎪⎨ 2 y + b1 − y x 1 2x 2 ]+ d1,1 − y 1 y x 11,1 +4 + 2b x 1 x 1 8 + 4b x 1 x 1+ (y 1,1 ) 2 R + (y 1 ) 3 R + (y 1 ) 2 y 1,1 R,[ ] [ ]⎪⎩ a 2 1d= y 1,1 − y 1 y x 1 x 11,1 + (y 1,1 ) 2 R + (y 1 ) 3 R + (y 1 ) 2 y 1,1 R.2 + b x 1 x 1 2(2 + b x 1 x 1)2

278We then get (notice the change of remainder):(8.65)a 1 a 1 = 1 [ 2x4 (y 1) 2 1 ] [+ b− y 1 y x 12x1,1 − (y 1 ) 2 2 ]+ dy x 11,1 + (y 1 ) 3 R + (y 1,1 ) 2 R,4 + 2b x 1 x 1 8 + 4b x 1 x[ ] [ ]1a 1 a 2 1= y 1 y 1,1 − (y 1 ) 2 dy x 1 x 11,1 + (y 1 ) 3 R + (y 1,1 ) 2 R,4 + 2b x 1 x 1 (4 + 2b x 1 x 1)2a 2 a 2 = (y 1 ) 3 R + (y 1,1 ) 2 R,[ 6xa 1 a 1 a 1 = −(y 1 ) 2 1 ]+ 3by x 11,1 (y 1 ) 3 R + (y 1,1 ) 2 R,16 + 8b x 1 x[ ]1a 1 a 1 a 2 = (y 1 ) 2 1y 1,1 + (y 1 ) 3 R + (y 1,1 ) 2 R.8 + 4b x 1 x 1Differentiating (8.60) with respect to x 2 , substituting y 2 for y x 2 and replacingd x 2 by b x 1 thanks to (8.61) 2 , we get(8.66)y 2 = a 1 a 1 + a 1 a 2 [2x 1 + b x 1] + a 2 a 2 [x 1 x 1 + e x 2] + a 1 a 1 a 1 [f x 2] + a 1 a 1 a 2 [2x 2 + g x 2]++ (a 1 ) 4 R + (a 1 ) 3 a 2 R + a 1 (a 2 ) 2 R + (a 2 ) 3 R.Replacing the monomials (8.65), we finally obtain:(8.67)y 2 = 1 [ ]4 (y 1) 2 + (y 1 ) 2 2gx 2 − 2d x 1 − (6x 1 + 3b x 1)f x 2y 1,1 − (2x1 + b x 1)d x 1 x 1+16 + 8b x 1 x 1 (4 + 2b x 1 x 1)2+ (y 1 ) 3 R + (y 1,1 ) 2 R.Thanks to (8.61) 4 , the first (big) coefficient of (y 1 ) 2 y 1,1 is zero; then theremainder coefficient is an O(x 1 ), hence vanishes at x = 0, together with itspartial first derivative with respect to x 2 . Accordingly, by s ∗ = s ∗ (x 1 , x 2 ),we will denote an unspecified function satisfying(8.68) s ∗ (0) = 0 and s ∗ x 2(0) = 0 .Lemma 8.69. The skeleton of the PDE system (E 5 ) associated to the submanifold(8.58) possesses three main equations of the form(∆ ⎧ E5 )y 2 = 1 4 (y 1) 2 + (y 1 ) 3 r + (y 1 ) 4 r + (y 1 ) 5 r + (y 1 ) 6 R+⎪⎨+ y 1,1[(y1 ) 2 s ∗ + (y 1 ) 3 r + (y 1 ) 4 r + (y 1 ) 5 r ] + (y 1,1 ) 2 R,y 1,2 = 1 2 y 1y 1,1 + (y 1 ) 3 r + (y 1 ) 4 r + (y 1 ) 5 r + (y 1 ) 6 R+[+ y 1,1 (y1 ) 2 r + (y 1 ) 3 r + (y 1 ) 4 r + (y 1 ) 5 r ] + (y 1 ) 6 R,[y 1,1,1 = (y 1 ) 3 r + (y 1 ) 4 r + y 1,1 r + y1 r + (y 1 ) 2 r + (y 1 ) 3 r ] +⎪⎩+ (y 1,1 ) 2[ r + y 1 r + (y 1 ) 2 r + (y 1 ) 3 r ] + (y 1,1 ) 3 R,

where the letter r denotes an unspecified function of (x 1 , x 2 ), and where thecoefficient s ∗ of (y 1 ) 2 y 1,1 in the first equation satisfies (8.68).Proof. To get the second equation, we compute:279(8.70)y 1,2 = a 1 a 2 [2 + b x 1 x 1] + a1 a 1 a 2 [g x 1 x 2] + (a1 ) 3 R + (a 2 ) 2 R= 1 2 y 1y 1,1 + (y 1 ) 2 y 1,1 r + (y 1 ) 3 R + (y 1,1 ) 2 R.The third equation is got similarly from (8.63) 3 . To conclude, we developethe first two equations mod [ (y 1 ) 6 , (y 1,1 ) 2] and the third onemod [ (y 1 ) 4 , (y 1,1 ) 3] .This precise skeleton will be referred to as ∆ E5 in the sequel. With theletter r, the computation rules are cst.r = r + r = r + s ∗ = r · r = r;sometimes, s ∗ may be replaced plainly by r.8.71. Infinitesimal Lie symmetries of (E 5 ). Letting L = X 1 ∂X 2 ∂ + Y ∂ be a candidate infinitesimal Lie symmetry and applying∂x 2 ∂y(8.72)L (3) = X 1∂∂x + X 2 ∂1 ∂x + Y ∂2 ∂y + Y 1+ Y 1,1∂∂y 1,1+ Y 1,2∂∂y 1,2+ Y 2,1+ Y 1,1,1∂∂y 1,1,1+ · · · + Y 2,2,2∂+ Y 2∂y 1∂+ Y 2,2∂y 2,1∂∂y 2,2,2to the skeleton ∆ E5 , we obtain firstly, computing mod [ (y 1 ) 5 , y 1,1]:0 ≡ −Y 2 + 1 2 y 1 Y 1 +∂∂y 2+∂x 1 +∂∂y 2,2+(8.73)+ (y 1 ) 3 r X 1 + (y 1 ) 4 r X 1 + (y 1 ) 3 r X 2 + (y 1 ) 4 r X 2 ++ Y 1[(y1 ) 2 r + (y 1 ) 3 r + (y 1 ) 4 r ] ++ Y 1,1[(y1 ) 2 s ∗ + (y 1 ) 3 r + (y 1 ) 4 r ] ,secondly, computing mod [ (y 1 ) 5 , y 1,1]:0 ≡ −Y 1,2 + 1 2 y 1 Y 1,1 +(8.74)+ (y 1 ) 3 r X 1 + (y 1 ) 4 r X 1 + (y 1 ) 3 r X 2 + (y 1 ) 4 r X 2 ++ Y 1[(y1 ) 2 r + (y 1 ) 3 r + (y 1 ) 4 r ] ++ Y 1,1[(y1 ) 2 r + (y 1 ) 3 r + (y 1 ) 4 r ] ,

280and thirdly, computing mod [ (y 1 ) 3 , (y 1,1 ) 2] :(8.75)0 ≡ −Y 1,1,1 + y 1,1 X 1 + y 1,1 X 2 ++ y 1,1 y 1 X 1 + y 1,1 y 1 X 2 + y 1,1 (y 1 ) 2 X 1 + y 1,1 (y 1 ) 2 X 2 ++ Y 1[(y1 ) 2 r ] + Y 1,1[r + y1 r + (y 1 ) 2 r ] + y 1,1 Y 1[r + y1 r + (y 1 ) 2 r ] ++ y 1,1 Y 1,1[r + y1 r + (y 1 ) 2 r ] .Specializing to n = 2 the formulas (3.9)(II), (3.20)(II) and (3.24)(II), we getY 1 , Y 2 , Y 1,1 , Y 1,2 and Y 1,1,1 :(8.76)Y 1 = Y x 1 + [ ]Y y −X 1x 1 y1 + [ ]−X 2x 1 y2 + [ ]−Xy1 (y1 ) 2 + [ ]−Xy2 y1 y 2 .(8.77)Y 2 = Y x 2 + [ ]− Xx 1 2 y1 + [ ]Y y − Xx 2 2 y2 + [ ]−Xy1 y1 y 2 + [ ]−Xy2 y2 y 2 .(8.78) ⎧Y 1,1 = Y x 1 x 1⎪⎨+ [ ]2 Y x 1 y − X 1x 1 x 1 y1 + [ ]− X 2x 1 x 1 y2 + [ Y yy − 2 X 1x y] 1 (y1 ) 2 ++ [ ]− 2 X 2x 1 y y1 y 2 + [ ]− Xyy1 (y1 ) 3 + [ − Xyy] 2 (y1 ) 2 y 2 ++ [ ]Y y − 2 X 1x⎪⎩1 y1,1 + [ ]− 2 X 2x 1 y1,2 + [ ]− 3 Xy1 y1 y 1,1 ++ [ ]− Xy2 y2 y 1,1 + [ ]− 2 X 2 y1 y 1,2 .y(8.79) ⎧Y 1,2 = Y x 1 x 2 + [ ]Y x 2 y − X 1x 1 x 2 y1 + [ ]Y x 1 y − X 2x 1 x 2 y2 +⎪⎨+ [ ]− X 1x 2 y (y1 ) 2 + [ Y yy − X 2x 2 y − X ] 1x 1 y y1 y 2 + [ − X 2x y] 1 y2 y 2 ++ [ ]− Xyy1 (y1 ) 2 y 2 + [ − Xyy] 2 y1 (y 2 ) 2 ++ [ ]− X 1x⎪⎩2 y1,1 + [ Y y − X 2x − X ] 1 2 x 1 y1,2 + [ ]− X 2x 1 y2,2 ++ [ ]− 2 Xy1 y1 y 1,2 + [ ]− 2 Xy2 y2 y 1,2 .

(8.80) ⎧Y 1,1,1 = Y x 1 x 1 x 1 + [ ]3 Y x 1 x 1 y − X 1x 1 x 1 x 1 y1 + [ ]− X 2x 1 x 1 x 1 y2 ++ [ ]3 Y x 1 yy − 3 X 1x 1 x 1 y (y1 ) 2 + [ − 3 X 2x 1 x y] 1 y1 y 2 ++ [ ]Y yyy − 3 Xx 1 1 yy (y1 ) 3 + [ − 3 Xx yy] 2 1 (y1 ) 2 y 2 ++ [ ]− Xyyy1 (y1 ) 4 + [ − Xyyy] 2 (y1 ) 3 y 2 +⎪⎨ + [ ]3 Y x 1 y − 3 X 1x 1 x 1 y1,1 + [ ]− 3 X 2x 1 x 1 y1,2 ++ [ ]3 Y yy − 9 X 1x 1 y y1 y 1,1 + [ − 3 X 2x y] 1 y2 y 1,1 ++ [ ]− 6 X 2x 1 y y1 y 1,2 + [ ]− 6 Xyy1 (y1 ) 2 y 1,1 + [ − 3 Xyy] 2 y1 y 2 y 1,1 ++ [ ]− 3 Xyy2 (y1 ) 2 y 1,2 + [ ]− 3 Xy1 (y1,1 ) 2 + [ ]− 3 Xy2 y1,1 y 1,2 ++ [ ]Y y − 3 X 1x⎪⎩1 y1,1,1 + [ ]− 3 X 2x 1 y1,1,2 + [ ]− 4 Xy1 y1 y 1,1,1 ++ [ ]− Xy2 y2 y 1,1,1 + [ ]− 3 X 2 y1 y 1,1,2 .y281Inserting Y 2 , Y 1,2 , Y 1,1,1 , Y 1 , Y 1,1 in the three Lie equations (8.73), (8.74),(8.75), replacing y 2 , y 1,2 , y 1,1,1 by the values they have on ∆ E5 , we get firstlyfive linear PDEs by picking the coefficients of cst., of y 1 , of (y 1 ) 2 , of (y 1 ) 3 ,of (y 1 ) 4 in (8.73):(8.81) ⎧0 = Y x 2,⎪⎩0 = X 1x 2 + 1 2 Y x 1,⎪⎨ 0 = Y y + X 2x − 2 X 1 2 x + r Y 1 x 1 + s∗ Y x 1 x 1,0 = 2 Xy 1 + X 2x + r X 1 + r X 2 + r Y 1 x 1 + r Y y + r X 1x 1++ r Y x 1 x 1 + s∗ Y x 1 y + s ∗ X 1x 1 x 1,0 = Xy 2 + r X 1 + r X 2 + r Y x 1 + r Y y + r X 1x + r X 2 1 x + r X 1 1 y ++ r Y x 1 x 1 + r Y x 1 y + r X 1x 1 x 1 + s∗ X 2x 1 x 1 + s∗ Y yy + s ∗ X 1x 1 y ,secondly, we get three more linear PDEs by picking the coefficients of (y 1 ) 2 ,of (y 1 ) 3 , of (y 1 ) 4 in (8.74):(8.82) ⎧0 = 3 Y x 1 y + X 2x 1 x + 4 X 1 2 x 2 y − 2 X 1x 1 x + r X 1 + r X 2 + r Y 1 x 1 + r Y x 1 x 1,⎪⎨0 = 2 Y yy + 2 X 2x 2 y − 6 X 1x 1 y − X 2x 1 x + r X 1 + r X 2 + r Y 1 x 1 + r Y y + r X 1x 1++ r Y x 1 x 1 + r Y x 1 y + r X 1x 1 x 1,0 = 4 Xyy 1 + 3 X 2x 1 y + r X 1 + r X 2 + r Y x 1 + r Y y + r X 1x + r X 2 1 x + r X 1 1 y +⎪⎩+ r Y x 1 x 1 + r Y x 1 y + r X 1x 1 x 1 + r X 2x 1 x 1 + r Y yy + r X 1x 1 y .

282and thirdly, we get five more linear PDEs by picking the coefficients of cst.,of y 1 , of y 1,1 , of y 1 y 1,1 , of (y 1 ) 2 y 1,1 in (8.75)(8.83)⎧0 = Y x 1 x 1 x 1 + r Y x 1 x 1,⎪⎨⎪⎩0 = −3 Y x 1 x 1 y + X 1x 1 x 1 x + r Y 1 x 1 x 1 + r Y x 1 y + r X 1x 1 x 1,0 = Y x 1 y − Xx 1 1 x + r X 1 + r X 2 + r Y 1 x 1 + r Y y + r Xx 1 + r Y 1 x 1 x 10 = − 3 2 X 2x 1 x 1 + 3 Y yy − 9 X 1x 1 y + r X 1 + r X 2 + r X 1x 1 + r Y x 1 + r Y y++ r X 2x 1 + r X 1y + r Y x 1 x 1 + r Y x 1 y + r X 1x 1 x 1,0 = 6 X 1yy + 15 4 X 2x 1 y + r X 1 + r X 2 + r X 1x 1 + r Y x 1 + r Y y + r X 2x 1 + r X 1y + r X 2y ++ r Y x 1 x 1 + r Y x 1 y + r X 1x 1 x 1 + r X 2x 1 x 1 + r Y yy + r X 1x 1 y .Proposition 8.84. Setting as initial conditions the ten specific differentialcoefficients(8.85)P := P ( X 1 , X 2 , Y , Xy 1 , X 2x 2, Y x 1, Y y, X 2x 1 x 2, Y x 1 x 1, Y )yy= r X 1 + r X 2 + r Y + r Xy 1 + r X 2x + r Y 2 x 1 + r Y y + r X 2x 1 x + r Y 2 x 1 x 1 + r Y yy,it follows by cross differentiations and by linear substitutions from theLie equations (8.81) i , i = 1, 2, 3, 4, 5, (8.82) j , j = 1, 2, 3, (8.83) i , i =1, 2, 3, 4, 5, that X 1x, X 2 1 x, Y 1 x 2, X 1x, X 2 2 y , X 1x 1 y , X 2x 2 x, Y 2 x 1 x 2, X 1x 2 y ,X 2x 2 y , Y x 1 y, Xyy, 1 Y x 2 y, X 2x 1 x 1 x, Y 2 x 1 x 1 x 1, X 2x 1 x 2 xY 2 x 1 x 1 x 2, X 2x 1 x 2 y , Y x 1 x 1 y,Y x 1 yy, Y x 2 yy, Y yyy are uniquely determined as linear combinations of

(X 1 , X 2 , Y , Xy 1,X 2x, Y 2 x 1, Y y , X 2x 1 x, Y 1 x 1 x 1, Y yy), namely:⎧X 1x = 1 P, X 2 2 31 x = P, Y 1 x 2 = P,(8.86)⎪⎨⎪⎩X 1x = 4 P, X 2 52 y = P,X 1x 1 yX 1x 2 y6= P, X 2x 2 x 2 7 = P, Y x 1 x 2 8 = P,9= P, X 2 1011x 2 y = P, Y x 1 y = P,Xyy1 1213= P, Y x 2 y = P,X 2x 1 x 1 x 2 14= P, Y x 1 x 1 x 1 15= P,X 2x 1 x 2 x 2 16= P, Y x 1 x 1 x 2 17= P,X 2 1819x 1 x 2 y = P, Y x 1 x 1 y = P,Y x 1 yyY x 2 yy20= P,21= P,Y yyy22= P.Then the expressions P are stable under differentiation with respectto x 1 , to x 2 , to y and moreover, all other, higher orderpartial derivatives of X 1 , of X 2 , of Y may be expressed asP ( X 1 , X 2 , Y , Xy 1,X 2x, Y 2 x 1, Y y , X 2x 1 x, Y 2 x 1 x 1, Y yy).Corollary 8.87. Every infinitesimal Lie symmetry of the PDE system (E 5 ) isuniquely determined by the ten initial Taylor coefficients(8.88)X 1 (0), X 2 (0), Y (0), Xy 1 (0), X 2x 2(0), Y x 1(0), Y y(0), X 2x 1 x 2(0), Y x 1 x 1(0), Y yy(0).Proof of the proposition. At first, (8.83) 1 yields (8.86) 15 ; (8.81) 1 yields(8.86) 3 ; differentiating (8.81) 1 with respect to x 1 yields (8.86) 8 ; differentiating(8.81) 1 with respect to y yields (8.86) 13 ; differentiating (8.81) 1 withrespect to x 1 x 1 yields (8.86) 17 ; and differentiating (8.81) 1 with respect to yyyields (8.86) 21 . Also, rewriting (8.81) 2 as(8.89) X 1x 2 = −1 2 Y x 1,we get (8.86) 4 ; and rewriting (8.81) 3 as(8.90) X 1x 1 = 1 2 X 2x 2 + 1 2 Y y + r Y x 1 + s ∗ Y x 1 x 1,we get (8.86) 1 .Next, differentiating (8.81) 2 with respect to x 1 and (8.81) 3 with respect tox 2 , we get, taking account of 0 = Y x 2 y = Y x 1 x 2 = Y x 1 x 1 x 2, replacing X x 1 x 2283

284by − 1 Y 2 x 1 2x1 and solving for Xx 2 x: 2(8.91)0 = X 1x 1 x 2 + 1 2 Y x 1 x 1,X 2x 2 x 2 = −(1 + s∗ x 2) Y x 1 x 1 + r Y x 1.This is (8.86) 7 . Differentiating (8.91) 2 with respect to x 1 , taking account of(8.83) 1 , we get (8.86) 16 :(8.92) X 2x 1 x 2 x 2 = r Y x 1 + r Y x 1 x 1.We then replace X 1x 1 from (8.90) in (8.81) 4 :(8.93)0 = X 2x 1 + 2 X 1y + r X 1 + r X 2 + r X 2x 2 + r Y x 1 + r Y y++ r Y x 1 x 1 + s∗ Y x 1 y + s ∗ X 1x 1 x 1.We differentiate this equation with respect to x 2 , knowing Y x 2 = 0:(8.94)0 = Xx 2 1 x + 2 X 1 2 x 2 y + r X 1 + r Xx 1 + r X 2 + r X 22 x + r X 2 2 x 2 x + r Y 2 x 1 + r Y y++ r Y x 1 x 1 + s∗ x 2 Y x 1 y + s ∗ x 2 X 1x 1 x 1 + s∗ X x 1 x 1 x 2.We replace: X 1xfrom (8.89); X 22 x 2 xfrom (8.91) 2 2 ; we differentiate (8.81) 2with respect to x 1 x 1 to replace X 1x 1 x 1 xby r Y 2 x 1 x 1, thanks to (8.83) 1; and wereorganize:(8.95)2 X 1x 2 y +s∗ x Y 2 x 1 y+s ∗ x X 1 2 x 1 x = −X 2 1 x 1 x 2+r X 1 +r X 2 +r X 2x 2+r Y x 1+r Y y+r Y x 1 x 1.We differentiate (8.81) 2 with respect to y and (8.81) 3 with respect to x 1 :(8.96)X 1x 2 y + 1 2 Y x 1 y = 0,Y x 1 y − 2 X 1x 1 x 1 = −X 2x 1 x 2 + r Y x 1 + r Y x 1 x 1.For the three unknowns X 1x 1 x, Y 1 x 1 y, X 1x 2 y, we solve the three equations(8.95), (8.96) 1 , (8.96) 2 , reminding s ∗ x(0) = 0: 2(8.97)X 1x 1 x = r X 1 + r X 2 + r X 21 x + r Y 2 x 1 + r Y y + r Y x 1 x 1 + r X 2x 1 x 2,Y x 1 y = r X 1 + r X 2 + r X 2x + r Y 2 x 1 + r Y y + r Y x 1 x 1 + r X 2x 1 x 2,X 1x 2 y = r X 1 + r X 2 + r X 2x 2 + r Y x 1 + r Y y + r Y x 1 x 1 + r X 2x 1 x 2.We get (8.86) 11 and (8.86) 9 .Thus, we may replace X 1x 1 x 1 and Y x 1 y in (8.81) 4 to get (8.86) 2 :(8.98)X 2x 1 = −2 X 1y + r X 1 + r X 2 + r X 2x 2 + r Y x 1 + r Y y + r Y x 1 x 1 + r X 2x 1 x 2.Next, we differentiate (8.83) 3 with respect to x 1 and we replace: X 1xfrom 1(8.90); X 2xfrom (8.98); Y 1 x 1 y from (8.97) 2 ; X 1x 1 xfrom (8.97) 1 1 ; Y x 1 x 1 x 1from (8.83) 1 ; and we compare with (8.83) 2 ; we differentiate (8.96) 1 with

espect to x 1 and (8.96) 2 with respect to x 1 ; solving, we obtain four newrelations:(8.99)X 1x 1 x 1 x = r X 1 + r X 2 + r X 21 x + r Y 2 x 1 + r Y y + r Y x 1 x 1 + r X 2x 1 x 2,Y x 1 x 1 y = r X 1 + r X 2 + r X 2x + r Y 2 x 1 + r Y y + r Y x 1 x 1 + r X 2x 1 x 2,X 2x 1 x 2 y = r X 1 + r X 2 + r X 2x + r Y 2 x 1 + r Y y + r Y x 1 x 1 + r X 2x 1 x 2,X 2x 1 x 1 x = r X 1 + r X 2 + r X 22 x + r Y 2 x 1 + r Y y + r Y x 1 x 1 + r X 2x 1 x 2.We get (8.86) 19 and (8.86) 14 .Next, in (8.81) 5 , we replace: X 1xfrom (8.90); X 21 xfrom (8.98); Y 1 x 1 yfrom (8.97) 2 ; we get:(8.100)Xy 2 = r X 1 + r X 2 + r Xy 1 + r X 2x + r Y 2 x 1 + r Y y + r Y x 1 x 1++ s ∗ X 2x 1 x + 1 s∗ Y yy + s ∗ X 1x 1 y .We differentiate (8.98) with respect to x 1 and we replace: X 1xfrom (8.90);1X 2xfrom (8.98); Y 1 x 1 y from (8.97) 2 ; Y x 1 x 1 x 1 from (8.83) 1; X 2x 1 x 1 xfrom 2(8.99) 4 ; we get:(8.101)X 2x 1 x 1+2 X 1x 1 y = r X 1 +r X 2 +r Xy 1 +r X 2x 2+r Y x 1+r Y y+r Y x 1 x 1+r X 2x 1 x 2.In (8.82) 2 , we replace: X 1xfrom (8.90); X 11 x 1 xfrom (8.97) 1 1 ; Y x 1 y from(8.97) 2 ; and we reorganize:(8.102)2 Xx 2 2 y −6 X x 1 1 y −X x 2 1 x = −2 Y yy+r X 1 +r X 2 +r X 21 x 2+r Y x 1+r Y y+r Y x 1 x 1+r X x 2 1 x 2.Differentiating (8.81) 3 with respect to y, we replace: Y x 1 y from (8.97) 2 ;Y x 1 x 1 y from (8.99) 2 ; and we reorganize:(8.103)X 2x 2 y −2 X 1x 1 y = −Y yy+r X 1 +r X 2 +r X 2x 2+r Y x 1+r Y y+r Y x 1 x 1+r X 2x 1 x 2.For the three unknowns X 2x 1 x, X 1 1 x 1 y , X 2x 2 y, we then solve the four equations(8.101), (8.102), (8.103), (8.83) 4 (in which we replace: X 1xfrom (8.90);1X 2xfrom (8.98); Y 1 x 1 y from (8.97) 2 ; X 1x 1 xfrom (8.97) 1 1 ):(8.104)X 2x 1 x = r X 1 + r X 2 + r X 11 y + r X 2x + r Y 2 x 1 + r Y y + r Y x 1 x 1 + r X 2x 1 x + r Y yy, 2X 1x 1 y = r X 1 + r X 2 + r Xy 1 + r X 2x + r Y 2 x 1 + r Y y + r Y x 1 x 1 + r X 2x 1 x + r Y yy, 2X 2x 2 y = r X 1 + r X 2 + r Xy 1 + r X 2x + r Y 2 x 1 + r Y y + r Y x 1 x 1 + r X 2x 1 x + r Y yy. 2We get (8.86) 6 and (8.86) 10 . Replacing then X 2x 1 x, X 1 1 x 1 yin (8.100) gives(8.105)Xy 2 = r X 1 +r X 2 +r Xy 1 +r X 2x 2+r Y x 1+r Y y+r Y x 1 x 1+r X 2x 1 x 2+r Y yy.This is (8.86) 5 .285

286Next, we differentiate (8.103) with respect to x 1 and we replace: X 1x 1from (8.90); X 2xfrom (8.98); Y 1 x 1 y from (8.97) 2 ; Y x 1 x 1 x 1 from (8.83) 1;X 2x 1 x 1 xfrom (8.99) 2 4 ; we get:(8.106)Y x 1 yy+X 2x 1 x 2 y −2 X 1x 1 x 1 y = r X 1 +r X 2 +r X 2x 2+r Y x 1+r Y y+r Y x 1 x 1+r X 2x 1 x 2.Also, we differentiate (8.83) 3 with respect to y and we replace: Xy 2 from(8.105); Y x 1 y from (8.97) 2 ; X 1x 1 y from (8.104) 2; Y x 1 x 1 y from (8.99) 2 ; weget:(8.107)Y x 1 yy−X 1x 1 x 1 y = r X 1 +r X 2 +r Xy 1 +r X 2x 2+r Y x 1+r Y y+r Y x 1 x 1+r X 2x 1 x 2+r Y yy.Also, we replace in (8.82) 3 : X 1xfrom (8.90); X 21 xfrom (8.98); Y 1 x 1 y from(8.97) 2 ; X 1x 1 xfrom (8.97) 1 1 ; X 2x 1 xfrom (8.104) 1 1 ; X 1x 1 y from (8.104) 2; weget:(8.108)4 Xyy 1 +3 X 2x 1 y = r X 1 +r X 2 +r Xy 1 +r X 2x 2+r Y x 1+r Y y+r Y x 1 x 1+r X 2x 1 x 2+r Y yy.We differentiate this equation with respect to x 2 and we replace: 4 X 1x 2 yyby −2 Y x 1 yy from (8.89); X 1xfrom (8.98); X 12 x 2 y from (8.97) 3; (notice 0 =Y x 1 x 2 = Y x 2 y); X 2x 2 xfrom (8.91) 2 2 ; X 2x 1 x 2 xfrom (8.92); we get:2(8.109)−2 Y x 1 yy+3 X 2x 1 x 2 y = r X 1 +r X 2 +r Xy 1 +r X 2x 2+r Y x 1+r Y y+r Y x 1 x 1+r X 2x 1 x 2+r Y yy.For the three unknowns X 1x 1 x 1 y , Y x 1 yy, X 2x 1 x 2 y, we solve the three equations(8.106), (8.107), (8.108); we get:(8.110)X 1x 1 x 1 y = r X 1 + r X 2 + r Xy 1 + r X 2x + r Y 2 x 1 + r Y y + r Y x 1 x 1 + r X 2x 1 x + r Y yy, 2Y x 1 yy = r X 1 + r X 2 + r Xy 1 + r X 2x + r Y 2 x 1 + r Y y + r Y x 1 x 1 + r X 2x 1 x + r Y yy, 2X 2x 1 x 2 y = r X 1 + r X 2 + r Xy 1 + r X 2x + r Y 2 x 1 + r Y y + r Y x 1 x 1 + r X 2x 1 x + r Y yy. 2We get (8.86) 20 and (8.86) 18 .Next, in (8.93), we replace: Y x 1 y from (8.97) 2 ; X 1x 1 xfrom (8.97) 1 1 ; weget:(8.111)X 2x + 2 X 1 1 y = r X 1 + r X 2 + r X 2x + r Y 2 x 1 + r Y y + r Y x 1 x 1 + r X 2x 1 x 2.We differentiate this equation with respect to y and we replace: Xy 2 from(8.105); X 2x 2 y from (8.104) 3; Y x 1 y from (8.97) 2 ; Y x 1 x 1 y from (8.99) 2 ; X 2x 1 x 2 yfrom (8.99) 3 ; we get:(8.112)X 2x 1 y +2 X yy 1 = r X 1 +r X 2 +r Xy 1 +r X 2x 2+r Y x 1+r Y y+r Y x 1 x 1+r X 2x 1 x 2+r Y yy.

For the two unknowns Xyy 1 and X 2x 1 y, we solve the two equations (8.108)and (8.112); we get:(8.113)Xyy 1 = r X 1 + r X 2 + r Xy 1 + r X 2x + r Y 2 x 1 + r Y y + r Y x 1 x 1 + r X 2x 1 x + r Y yy, 2X 2x 1 y = r X 1 + r X 2 + r Xy 1 + r X 2x + r Y 2 x 1 + r Y y + r Y x 1 x 1 + r X 2x 1 x + r Y yy. 2287We get (8.86) 12 .Next, we differentiate (8.113) 1 with respect to x 1 and we replace: X 1x, 1X 2x, X 1 1 x 1 y , Y x 1 y, Y x 1 x 1 x 1, X 2x 1 x 1 x, Y 2 x 1 yy; we get:(8.114)X 1x 1 yy = r X 1 +r X 2 +r Xy 1 +r X 2x 2+r Y x 1+r Y y+r Y x 1 x 1+r X 2x 1 x 2+r Y yy.Also, we differentiate (8.113) 2 with respect to x 1 and we replace:(8.115)X 2x 1 x 1 y = r X 1 +r X 2 +r Xy 1 +r X 2x 2+r Y x 1+r Y y+r Y x 1 x 1+r X 2x 1 x 2+r Y yy.Also, we differentiate (8.83) 4 with respect to y; we replace X 1x 1 yy from(8.114), we replace X 2x 1 x 1 yfrom (8.115); and we achieve other evident replacements;we get:(8.116)Y yyy = r X 1 +r X 2 +r Xy 1 +r X 2x 2+r Y x 1+r Y y+r Y x 1 x 1+r X 2x 1 x 2+r Y yy.This is (8.96) 22 , which completes the proof.Theorem 8.117. The bound dimSYM(E 5 ) 10 is attained if and only if(E 5 ) is equivalent, through a diffeomorphism (x 1 , x 2 , y) ↦−→ (X 1 , X 2 , Y ),to the model system(8.118) Y X 2 = 0, Y X 1 X 1 X 1 = 0.Proof. Firstly, setting r = s ∗ = 0 everywhere, the solution to (8.81), (8.82),(8.83) is(8.119)X 1 = k + (c + j) x 1 − bx 2 − h y + e x 1 x 1 − d x 1 x 2 + f x 1 y − e x 2 y,X 2 = g + 2h x 1 + 2j x 2 − d x 2 x 2 + 2e x 1 x 2 − f x 1 x 1 ,Y = a + 2bx 1 + 2c y + d x 1 x 1 + 2e x 1 y + f yy,

288where a, b, c, d, e, f, g, h, j, k ∈ K are arbitrary. Computing the third prolongationsof the ten vector fields(8.120)∂∂x 1,∂ ∂∂x 2, ∂y ,− x 2 ∂∂x 1 + ∂ 2x1 ∂y , ∂x1− x 1 x 2 ∂∂x 1 − x2 x 2x 1 y ∂∂x 1 − x1 x 1 ∂∂x 2 + yy ∂ ∂y∂x 1 + 2y ∂ ∂y , ∂x1∂x 1 + 2x2∂∂x 2 + x1 x 1 ∂ ∂y ,∂∂x 2,−y ∂∂x 1 + ∂2x1 ∂x 2,(x1 x 1 − x 2 y) ∂∂x 1 + 2x1 x 2 ∂∂x 2 + 2x1 y ∂ ∂y ,one verifies that they all are tangent to the skeleton y 2 = 1 4 (y 1) 2 , y 1,1,1 = 0.Thus the bound is attained. One then verifies ([FK2005a]) that the spannedLie algebra is isomorphic to so(5, C).Lemma 8.121. Assuming the normalizations of Lemma 8.54, the remainderO 4 in (8.53) is an O 3 (x 1 , a 1 ):(8.122)y = b+ 2 x1 a 1 + x 1 x 1 a 2 + a 1 a 1 x 21 − x 2 a 2 +(x 1 ) 3 R+(x 1 ) 2 a 1 R+x 1 (a 1 ) 2 R+(a 1 ) 3 R.Proof. Indeed, writing(8.123)y = b + x 1 Λ 1,0 + a 1 Λ 0,1 + x 1 x 1 Λ 2,0 + x 1 a 1 Λ 1,1 + a 1 a 1 Λ 0,2 + O 3 (x 1 , a 1 ),with Λ i,j = Λ i,j (x 2 , a 2 ), and developing the determinant (8.54) with respectto the powers of (x 1 , a 1 ), the vanishing of the coefficients of cst., of x 1 , ofa 1 yields the system(8.124)⎧⎪⎨⎪⎩0 ≡ Λ 1,0a 2 Λ 0,1x 2 ,0 ≡ Λ 1,1 Λ 1,0x 2 a 2 − 2 Λ 2,0a 2 Λ 0,1x 2 − Λ 1,1x 2 Λ 1,0a 2 ,0 ≡ Λ 1,1 Λ 0,1x 2 a 2 − Λ 1,1a 2 Λ 0,1x 2 − 2 Λ 0,2x 2 Λ 1,0a 2 .If the first equation yields Λ 1,0a≡ 0, replacing in the second, using Λ 2,0 =2a 2 +O 2 , we deduce that Λ 0,1x≡ 0 also. Similarly, Λ 0,12 x≡ 0 implies Λ 1,02 a≡ 0. 2Since the coordinate system satisfies the normalization Π(0, a) ≡ Π(x, 0) ≡0, necessarily Λ 1,0 = O(a 2 ) and Λ 0,1 = O(x 2 ). We deduce:(8.125) 0 ≡ Λ 1,0 ≡ Λ 0,1 .

Redeveloping the determinant, the vanishing of the coefficients of x 1 x 1 , ofx 1 a 1 , of a 1 a 1 yields the system⎧0 ≡ Λ ⎪⎨1,1 Λ 2,0x 2 a− 2 Λ 2,02 aΛ 1,12 x, 2(8.126) 0 ≡ Λ 1,1 Λ 1,1x⎪⎩2 a− Λ 1,12 aΛ 1,12 x− 4 Λ 2,02 aΛ 0,22 x, 20 ≡ Λ 1,1 Λ 0,2x 2 a− 2 Λ 0,22 xΛ 1,12 a. 2Since Λ 1,1 (0) = 2 ≠ 0, we may divide by Λ 1,1 , obtaining a PDE system withthe three functions Λ 2,0x 2 a, Λ 1,12 x 2 a, Λ 0,22 x 2 ain the left hand side. We observe that2the normalizations of Lemma 8.55 entail(8.127)Λ 2,0 = a 2 +O(x 2 a 2 ), Λ 1,1 = 2+O(x 2 a 2 ), Λ 0,2 = x 2 +O(x 2 a 2 ).By cross differentiations in the mentioned PDE system, it follows that all theTaylor coefficients of Λ 2,0 , Λ 1,1 , Λ 0,2 are uniquely determined. As alreadydiscovered in [GM2003b], the unique solution(8.128) Λ 2,0 a 22x 2=1 − x 2 a 2, Λ1,1 =1 − x 2 a 2, Λ0,2 =1 − x 2 a 2,guarantees, when the remainder O 3 (x 1 , a 1 ) vanishes, that the determinant(8.45) indeed vanishes identically.Conversely, suppose that dim SYM(E 5 ) = 10 is maximal.With ε ≠ 0 small, replacing (x 1 , x 2 , y, a 1 , a 2 , b) by(εx 1 , x 2 , ε 2 y, εa 1 , a 2 , εεb) in (8.122) and dividing by εε, the remainderterms become small:(8.129) y = b + 2 x1 a 1 + x 1 x 1 a 2 + a 1 a 1 x 21 − x 2 a 2 + O(ε).Then all the remainders in the equations ∆ E5 of the skeleton are O(ε). Weget ten generators similar to (8.120), plus an O(ε) perturbation. Thanks tothe rigidity of so(5, C), Theorem 5.15 provides a change of generators, closeto the 10×10 identity matrix, leading to the same structure constants as thoseof the ten vector fields (8.120). As in the end of the proof of Theorem 5.13,we may then straighten some relevant vector fields (exercise) and finallycheck that their tangency to the skeleton implies that it is the model one.Theorem 8.117 is proved.Corollary 8.130. Let M ⊂ C 3 be a connected real analytic hypersurfacewhose Levi form has uniform rank 1 that is 2-nondegenerate at every point.Then(8.131) dimhol(M) 10,and the bound is attained if and only if M is locally, in a neighborhood ofZariski-generic points, biholomorphic to the model M 0 .289

290§9. DUAL SYSTEM OF PARTIAL DIFFERENTIAL EQUATIONS9.1. Solvability with respect to the variables. Let M be as in §2.10 definedby y = Π(x, a, b) or dually by b = Π ∗ (a, x, y).Definition 9.2. M is solvable with respect to the variables if there exist aninteger κ ∗ 1 and multiindices δ(1), . . .,δ(n) ∈ N p with |δ(l)| 1 for l =1, . . ., n and max 1ln |δ(l)| = κ ∗ , together with integers j(1), . . .,j(n)with 1 j(l) m such that the local K-analytic map(9.3) ( (ΠK n+m ∋ (x, y) ↦−→ ∗j (0, x, y) ) ( ) )1jm, Π ∗ j(l)(0, x, y) ∈ K m+na δ(l) 1lnis of rank equal to n + m at (x, y) = (0, 0)If M is a complexified generic submanifold, solvability with respect tothe parameters is equivalent to solvability with respect to the variables, becauseΠ ∗ = Π. This is untrue in general: with n = 2, m = 1, consider thesystem y x 2 = 0, y x 1 x 1 = 0, whose general solutions is y(x) = b + x 1a withx 2 absent.To characterize generally such a degeneration property, we develope both(9.4)y j = Π j (x, a, b) = ∑ x β Π j β(a, b) andβ∈N nb j = Π ∗j (a, x, y) = ∑ a δ Π ∗j δ(x, y),δ∈N pwith analytic functions Π j β (a, b), Π∗ jδ (x, y) and we introduce two K∞ -valuedmaps(9.5)Q ∞ :Q ∗ ∞ :(a, b) ↦−→ ( Π j β (a, b)) 1jmβ∈N n(x, y) ↦−→ (Π ∗j δ (x, y)) 1jmδ∈N p .Since b ↦→ ( Π j 0(0, b) ) 1jmand y ↦→(Π∗j0(0, y) ) 1jmare already both ofrank m at the origin, the generic ranks of these two maps, defined by testingthe nonvanishing of minors of their infinite Jacobian matrices, satisfy(9.6)genrkQ ∞ = m + p MgenrkQ ∗ ∞ = m + n Mfor some two integers 0 p M p and 0 n M n. So at a Zariski-genericpoint, the ranks are equal to m + p M and to m + n M .Proposition 9.7. There exists a local proper K-analytic subset Σ M of K n x ×K m y × K p a × K m b whose equations, of the specific form{}(9.8) Σ M = r ν (a, b) = 0, ν ∈ N, rµ(x, ∗ y) = 0 µ ∈ N ,andand

291are obtained by equating to zero all (m+p M )×(m+p M ) minors of Jac Q ∞and all (m + n M ) × (m + n M ) minors of Jac Q∞ ∗ , such that for everypoint p = (x p , y p , a p , b p ) ∉ Σ M , there exists a local change of coordinatesrespecting the separation of the variables (x, y) and (a, b)(9.9) (x, y, a, b) ↦→ ( ϕ(x, y), h(a, b) ) =: (x ′ , y ′ , a ′ , b ′ )by which M is transformed to a submanifold M ′ centered and localized atp ′ = p having equations(9.10) y ′ = Π ′ (x ′ , a ′ , b ′ ) and dually b ′ = Π ′∗ (a ′ , x ′ , y ′ )with Π ′ and Π ′∗ independent of((9.11) x′nM +1 , . . ., n)x′and of(a′pM +1 , . . ) .,a′ p .So M ′ , may be considered to be living in K n Mx ′ × K m y ′ × Kp Ma ′ × K m b ′ andin such a smaller space, at p ′ = p, it is solvable both with respect to theparameters and to the variables.Interpretation: by forgetting some innocuous variables, at a Zariskigenericpoint, any M is both solvable with respect to the parameters andto the variables. These two assumptions will be held up to the end of thisPart I.9.12. Dual system (E ∗ ) and isomorphisms SYM(E ) ≃SYM ( V S (E ) ) = SYM ( V S (E ∗ ) ) ≃ SYM(E ∗ ). To a system(E ), we associate its submanifold of solutions M := V S (E ). Assumingit to be solvable with respect to the variables and proceeding as in §2.10,we can derive a dual system of completely integrable partial differentialequations of the form((E ∗ ) b j a γ(a) = Gj γ a, b(a), ( b j(l) (a) ) ),a δ(l) 1lnwhere (j, γ) ≠ (j, 0) and ≠ (j(l), δ(l)). Its submanifold of solutionsV S (E ∗ ) ≡ V S (E ) has equations dual to those of V S (E ).Theorem 9.13. Under the assumption of twin solvability, we have:(9.14) SYM(E ) ≃ SYM ( V S (E ) ) = SYM ( V S (E ∗ ) ) ≃ SYM(E ∗ ),through L ←→ L + L ∗ = L ∗ + L ←→ L ∗ .§10. FUNDAMENTAL PAIR OF FOLIATIONS AND COVERING PROPERTY10.1. Fundamental pair of foliations on M . As in §2, let (E ) and M =V S (E ) be defined by y = Π(x, a, b) or dually by b = Π ∗ (a, x, y). Abbreviate(10.2) z := (x, y) and c := (a, b).

292Every transformation (z, c) ↦→ ( ϕ(z), h(c) ) belonging to G v,p stabilizes both{z = cst.} and {c = cst.}. Accordingly, the two foliations of M(10.3) F v := ⋃ M ∩ { }c = c 0 and F p := ⋃ M ∩ { }z = z 0c 0 z 0are invariant under changes of coordinates. We call (F v , F p ) the fundamentalpair of foliations on M . The leaves of the foliation by variables F v are n-dimensional:(10.4) F v (c 0 ) = { (z, c 0 ) : y = Π(x, c 0 ) }The leaves of the foliation by parameters F p are p-dimensional:(10.5) F p (c 0 ) = { (z 0 , c) : b = Π ∗ (a, z 0 ) }We draw a diagram. In it, the positive codimension is invisible:(10.6) m = dimM − dimF v − dim F p 1cL ∗ 0F vF pzLMM10.7. Chains Γ k and dual chains Γ ∗ k . Similarly as in [GM2004, Me2005a,Me2005b, MP2005] (in a CR context), we introduce two collections(L k ) 1kn and (L ∗ q ) 1qp of vector fields whose integral manifolds coincidewith the leaves of F v and of F p :(10.8)⎧⎪⎨⎪⎩L k := ∂∂x k+m∑j=1L ∗ q := ∂∂a q + m∑j=1∂Π j∂x k(x, a, b) ∂∂y j ,k = 1, . . .,n,∂Π ∗j ∂(a, x, y) , q = 1, . . ., p.∂aq ∂bj

Let (z 0 , c 0 ) = (x 0 , y 0 , a 0 , b 0 ) ∈ M be a fixed point, let x 1 := (x 1 1, . . ., x n 1) ∈K n and define the multiple flow map(10.9) {Lx1 (x 0 , y 0 , a 0 , b 0 ) := exp(x 1 L)(p 0 ) := exp ( x n 1 L n(· · ·(exp(x11 L 1 (z 0 , c 0 ))) · · ·) ):= ( x 0 + x 1 , Π(x 0 + x 1 , a 0 , b 0 ), a 0 , b 0).Similarly, for a 1 = (a 1 1, . . ., a p 1) ∈ K p , define the multiple flow map(10.10) L ∗ a 1(x 0 , y 0 , a 0 , b 0 ) := ( x 0 , y 0 , a 0 + a 1 , Π ∗ (a 0 + a 1 , x 0 , y 0 ) ) .Starting from the (z 0 , c 0 ) = (0, 0) and moving alternately along F v , F p , F v ,etc., we obtain⎧Γ 1 (x 1 ) := L x1 (0),⎪⎨ Γ 2 (x 1 , a 1 ) := L ∗ a(10.11)1(L x1 (0)),Γ 3 (x 1 , a 1 , x 2 ) := L x2 (L⎪⎩∗ a 1(L x1 (0))),Γ 4 (x 1 , a 1 , x 2 , a 2 ) := L ∗ a 2(L x2 (L ∗ a 1(L x1 (0)))),and so on. Generally, we get chains Γ k := Γ k ([xa] k ), where [xa] k :=(x 1 , a 1 , x 2 , a 2 , . . .) with exactly k terms, where each x l ∈ K n and eacha l ∈ K p .If, instead, the first movement consists in moving along F p , we start withΓ ∗ 1 (a 1) := L ∗ a 1(0), Γ ∗ 2 (a 1, x 1 ) := L x1 (L ∗ a 1(0)), etc., and generally we get dualchains Γ ∗ k ([ax] k), where [ax] k := (a 1 , x 1 , a 2 , x 2 , . . .), with exactly k terms.Both Γ k and Γ ∗ k have range in M .For k = 1, 2, 3, · · · , integers e k and e ∗ k are defined inductively by{e1 + e 2 + e 3 + · · · + e k = genrk K (Γ k ),(10.12)e ∗ 1 + e∗ 2 + e∗ 3 + · · · + e∗ k = genrk K(Γ ∗ k ).By (10.9) and (10.10), it is clear that e 1 = n, e 2 = p, e ∗ 1 = p, and e∗ 2 = n.Example 10.13. For y xx = 0, the submanifold of solutions M is simplyy = b + xa, whence⎧⎪⎨Γ 1 (x 1 ) = (x 1 , 0, 0, 0),(10.14)Γ 2 (x 1 , a 1 ) = (x 1 , 0, a 1 , −x 1 a 1 ),⎪⎩Γ 3 (x 1 , a 1 , x 2 ) = (x 1 + x 2 , x 2 a 1 , a 1 , −x 1 a 1 ).The rank at (0, 0, 0) of Γ 3 is equal to two, not more. However, its genericrank is equal to three. Similar observations hold for the two submanifoldsof solutions y = b + xxa + xaa and y = b + xa 1 + xxa 2 (in K 5 ).Lemma 10.15. If genrk K (Γ k+1 ) = genrk K (Γ k ), then for each positive integerl 1, we have genrk K (Γ k+l ) = genrk K (Γ k ). The same stabilizationproperty holds for Γ ∗ k .293

29410.16. Covering property. We now formulate a central concept.Definition 10.17. The pair of foliations (F v , F p ) is covering at the originif there exists an integer k such that the generic rank of Γ k is (maximalpossible) equal to dim K M . Since for a 1 = 0, the dual (k + 1)-th chainΓ ∗ k+1 identifies with the k-th chain Γ k, the same property holds for the dualchains.Example 10.18. With n = 1, m = 2 and p = 1 the submanifold defined byy 1 = b 1 and y 2 = b 2 + xa is twin solvable, but its pair of foliations is notcovering at the origin. Then SYM(M ) is infinite-dimensional, since fora = a(y 1 ) an arbitrary function, it contains a(y 1 ) ∂ + a(b 1 ) ∂ .∂y 1 ∂b 1Because we aim only to study finite-dimensional Lie symmetry groups ofpartial differential equations, in the remainder of this Part I, we will constantlyassume the covering property to hold.By Lemma 10.15, there exist two well defined integers µ and µ ∗ suchthat e 3 , e 4 , . . .,e µ+1 > 0, but e µ+l = 0 for all l 2 and similarly,e ∗ 3 , e∗ 4 , . . .,e∗ µ ∗ +1 > 0, but e∗ µ ∗ +l = 0 for all l 2. Since the pair of foliationsis covering, we have the two dimension equalities{ n + p + e3 + · · · + e µ+1 = dim K M = n + m + p,(10.19)p + n + e ∗ 3 + · · · + e∗ µ ∗ +1 = dim K M = n + m + p.By definition, the ranges of Γ µ+1 and of Γ ∗ µ ∗ +1 cover (at least; more is true,see: Theorem 10.28) an open subset of M . Also, it is elementary to verifythe four inequalitiesµ 1 + m, µ ∗ 1 + m,(10.20)µ µ ∗ + 1, µ ∗ µ + 1.In fact, since Γ k+1 with x 1 = 0 identifies with Γ ∗ k , the second line follows.Definition 10.21. The type of the covering pair of foliations (F v , F p ) is thepair of integers(10.22) (µ, µ ∗ ), with max(µ, µ ∗ ) 1 + m.Example 10.23. (Continued) We write down the explicit expressions of Γ 4and of Γ 5 :(10.24)⎧⎪⎨Γ 4 (x 1 , a 1 , x 2 , a 2 ; 0) = ( x 1 + x 2 , x 2 a 1 , a 1 + a 2 , −x 1 a 1 − x 1 a 2 − x 2 a 2 , ) ,Γ 5 (x 1 , a 1 , x 2 , a 2 , x 3 ; 0) = ( x 1 + x 2 + x 3 , x 2 a 1 + x 3 a 1 + x 3 a 2 , a 1 + a 2⎪⎩)− x 1 a 1 − x 1 a 2 − x 2 a 2 .Here, dimM = 3. By computing its Jacobian matrix, Γ 5 is of rank 3 atevery point (x 1 , a 1 , 0, −a 1 , −x 1 ) ∈ K 5 with a 1 ≠ 0. Since (obviously)(10.25) Γ 5(x1 , a 1 , 0, −a 1 , −x 1)= 0 ∈ M,

we deduce that Γ 5 is submersive (“covering”) from a small neighborhood of(x1 , a 1 , 0, −a 1 , −x 1)in K 5 onto a neighborhood of the origin in M .10.26. Covering a neighborhood of the origin in( M . For (z 0 , c 0 ) ∈ Mfixed and close to the origin, we denote by Γ k [xa]k ; (z 0 , c 0 ) ) and byΓk( ∗ [ax]k ; (z 0 , c 0 ) ) the (dual) chains issued from (z 0 , c 0 ). For given parameters[xa] k = (x 1 , a 1 , x 2 , . . .), we denote by [−xa] k the collection(· · · , −x 2 , −a 1 , −x 2 ) with minus signs and reverse order; similarly, we introduce[−ax] k . Notably, we have L −x1 (L x1 (0)) = 0 (because L −x1 +x 1(·) =L 0 (·) = Id), and also L −x1 (L ∗ −a 1(L ∗ a 1(L x1 (0)))) = 0 and generally:(10.27) Γ k([−xa]k ; Γ k ([xa] k ; 0) ) ≡ 0.Geometrically speaking, by following backward the k-th chain Γ k , we comeback to 0.Theorem 10.28. ([Me2005a, Me2005b], [∗]) The two maps Γ 2µ+1 andΓ ∗ 2µ ∗ +1 are submersive onto a neighborhood of the origin in M . Precisely,there exist two points [xa] 0 2µ+1 ∈ K (µ+1)n+µp and [ax] 0 2µ ∗ +1 ∈K µ∗ n+(µ ∗ +1)p arbitrarily close to the origin with Γ 2µ+1 ([xa] 0 2µ+1) = 0 andΓ ∗ 2µ ∗ +1 ([ax]0 2µ ∗ +1 ) = 0 such that the two maps{ ( )K (µ+1)n+µp ∋ [xa] 2µ+1 ↦−→ Γ 2µ+1 [xa]2µ+1 ∈ M and(10.29)K µ∗ n+(µ ∗ +1)p ∋ [ax] 2µ ∗ +1 ↦−→ Γ ∗ 2µ ∗ +1([ax]2µ ∗ +1)∈ M295are of rank n + m + p = dim K M at the points [xa] 0 2µ and [ax] 0 2µ ∗ respectively.In particular, the ranges of the two maps Γ 2µ+1 and Γ ∗ 2µ ∗ +1 cover aneighborhood of the origin in M .Let π z (z, c) := z and π c (z, c) := c be the two canonical projections. Thenext corollary will be useful in Section 12. In the example above, it alsofollows that the map(10.30) [xa] 4 ↦→ π c(Γ4 ([xa 4 ]) ) = ( a 1 + a 2 , −x 1 a 1 − x 1 a 2 − x 2 a 2)∈ K2is of rank two at all points [xa] 0 4 := ( x 0 1 , a0 1 , 0, −a0 1)with a01 ≠ 0.Corollary 10.31. ([Me2005a, Me2005b], [∗]) There exist two points[xa] 0 2µ ∈ Kµ(n+p) and [ax] 0 2µ ∈ (n+p) arbitrarily close to the origin with( ∗ Kµ∗π c (Γ 2µ ([xa] 0 2µ)) = 0 and π z Γ∗2µ ∗([ax] 0 2µ ∗)) = 0 such that the two maps{ (K µ(n+p) ∋ [xa] 2µ ↦−→ π c Γ2µ ([xa] 2µ ) ) ∈ K m+p and(10.32)K µ∗ (n+p) ∋ [ax] 2µ ∗ ↦−→ π z(Γ∗2µ ∗([ax] 2µ ∗) ) ∈ K n+mare of rank m + p at the point [xa] 0 2µ ∈ Kµ(n+p) and of rank n + m at thepoint [ax] 0 2µ ∗ ∈ Kµ∗ (n+p) .

296In the case m = 1 (single dependent variable y ∈ K), the covering propertyalways hold with µ = µ ∗ = 2.§11. FORMAL AND SMOOTH EQUIVALENCES BETWEEN SUBMANIFOLDSOF SOLUTIONS11.1. Transformations of submanifolds of solutions. Lemma 7.3 showsthat every equivalence ϕ between two PDE systems (E ) and (E ′ ) lifts as atransformation which respects the separation between variables and parametersof the form(11.2)(x, y, a, b) ↦−→ ( φ(x, y), ψ(x, y), f(a, b), g(a, b) ) = ( ϕ(x, y), h(a, b) ) =: (x ′ , y ′ , a ′ , b ′ )from the source submanifolds of solutions M := V S (E ) to the targetM ′ := V S (E ′ ), whose equations are(11.3)y = Π(x, c) or dually b = Π ∗ (a, z) andy ′ = Π ′ (x ′ , c ′ ) or dually b ′ = Π ′∗ (a ′ , z ′ ).The study of transformations between submanifolds of solutions possessesstrong similarities with the study of CR mappings between CR manifolds([Pi1975, We1977, DW1980, BJT1985, DF1988, BER1999, Me2005a,Me2005b]). In fact, one may transfer the whole theory of the analytic reflectionprinciple to this more general context. In the present §10 and in the next§11, we select and establish some of the results that are useful to the Lie theory.Some accessible open questions will also be formulated.Maps of the form (11.2) send leaves of F v and of F p to leaves of F ′ v, andof F ′ p , respectively.MK n+2m+p 0F pK n+2m+pF ′ pcΓ ∗ ([ax] 3 )(ϕ, h)M ′(ϕ(z), h(c))F vΓ ∗ (a 1 )Γ ∗ ([ax] 2 )zF ′ vc ′ z ′0 ′

11.4. Regularity and jet parametrization. Some strong rigidity propertiesunderly ( ) the above ( diagram. ) Especially, the smoothness of the two pairsFv , F p and F′v , F ′ p governs the smoothness of (ϕ, h).(We shall)study the regularity of a purely formal map (z ′ , c ′ ) =ϕ(z), h(c) , namely ϕ(z) ∈ K[z] n+m and h(c) ∈ K[c] p+m , assuming (E )and (E ′ ) to be analytic. Concretely, the assumption that (ϕ, h) maps M toM ′ reads as one of the four equivalent identities:⎧ψ ( x, Π(x, c) ) ≡ Π ′( φ(x, Π(x, c)), h(c) ) ,⎪⎨ ψ(z) ≡ Π ′( φ(z), h(a, Π ∗ (a, z) ) ,(11.5)g ( a, Π ∗ (a, z) ) ≡ Π ′ ∗ ( f(a, Π ∗ (a, z)), ϕ(z) ) ,⎪⎩g(c) ≡ Π ′ ∗ ( f(c), ϕ(x, Π(x, c)) ) ,in K[x, c] m and in K[a, z] m .Theorem 11.6. Let (ϕ, h) := M → M ′ be a purely formal equivalencebetween two local K-analytic submanifolds of solutions. Assume that thefundamental pair of foliations (F v , F p ) is covering at the origin, with type(µ, µ ∗ ) at the origin. Assume that M ′ is both κ-solvable with respect to theparameters and κ ∗ -solvable with respect to the variables. Set l := µ ∗ (κ +κ ∗ ) and l ∗ := µ(κ ∗ + κ). Then there exist two K n+m -valued and K p+m -valued local K-analytic maps Φ l and H l ∗, constructible only by means ofΠ, Π ∗ , Π ′ , Π ′∗ , such that the following two formal power series identitieshold:{ (ϕ(z) ≡ Φl z, Jlz ϕ(0) ) ,(11.7)(h(c) ≡ H l ∗ c, Jl ∗c h(0) ) ,in K[z] n+m and in K[c] p+m , where Jz l ϕ(0) denotes the l-th jet of h at theorigin and similarly for Jc l∗ h(0). In particular, as a corollary, we have thefollowing two automatic regularity properties:• ϕ(z) ∈ K{z} n+m and h(c) ∈ K{c} p+m are in fact convergent;• if in addition M and M ′ are K-algebraic in the sense of Nash, thenΦ l and H l ∗ are also K-algebraic, whence ϕ(z) ∈ A K {z} n+m andh(c) ∈ A K {c} p+m are in fact K-algebraic.Proof. We remind the explicit expressions of the two collections of vectorfields spanning the leaves of the two foliations F v and F p :⎧L k := ∂ m∑ ∂Π j+ (x, c) ∂⎪⎨ ∂x k ∂x k ∂y , k = 1, . . .,n,j(11.8)⎪⎩j=1L ∗ q := ∂∂a q + m∑j=1∂Π ∗j ∂(a, z)∂aq ∂b , jq = 1, . . .,p.297

298Observe that differentiating the first line of (11.5) with respect to x k amountsto applying the derivation L k . Similarly, differentiating the third lineof (11.5) with respect to a q amounts to applying L ∗ q . We thus get for(z, c) ∈ M(11.9)⎧L ⎪⎨ k ψ(z) =⎪⎩n∑l=1L ∗ q g(c) = p∑r=1∂Π ′ ( )∂x ′ l φ(z), h(c) Lk φ l (z) and∂Π ′ ∗∂a ′ r(f(c), ϕ(z))L∗q f r (c),It follows from det ( ) (∂ϕ k∂z (0) ≠ 0 and det ∂h k) l ∂c (0) ≠ 0 that the two formalldeterminants(11.10) det ( L k φ l (z) ) 1ln1knand det ( L ∗ q fr (c) ) 1rp1qphave nonvanishing constant term. Consequently, these two matrices are invertiblein K[z] and in K[c]. So there exist universal polynomials S j landS ∗j r such that⎧(∂Π ′ j( ) S j {Lk′lϕ i′ (z) } )1i ′ n+m1k⎪⎨ ∂x ′ l ϕ(z), h(c) = ′ ndet ( L k ′ φ l′ (z) ) and1l ′ n1k(11.11)′ n( {L∂Π ′ ∗j( ) S ∗j ∗r q ′ h i′ (c) } )1i ′ p+m1q⎪⎩ ∂a ′ r f(c), ϕ(z) = ′ pdet ( L ∗ qf ′ r′ (c) ) ,1r ′ p1q ′ pfor 1 j m, for 1 l n, for 1 r p and for (z, c) ∈ M .Again, we apply the vector fields L k to the obtained first line and thevector fields L ∗ q to the obtained second line, getting, thanks to the chain rule:(11.12) ⎧( {Lkn∑ ∂ 2 Π ′ j( ) R j ′⎪⎨∂x ′ l 1x ′ l 2φ(z), h(c) Lk φ l l 1 ,k 1L k ′2ϕ i′ (z) } )1i ′ n+m21k 1(z) =′ ,k′ 2[nl 2 =1det ( L k ′ φ l′ (z) ) ]1l ′ n 2and1k ′ n⎪⎩p∑r 2 =1∂ 2 Π ′ ∗j∂a ′ r 1a ′ r 2(f(c), ϕ(z))L∗q f r 2(c) =( {LR ∗j ∗r 1 ,q q 1L ∗ ′ qh2′ i′ (c) } 1i ′ p+m[det ( L ∗ qf ′ r′ (c) ) ]1r ′ p 2,1q ′ p1q ′ 1 ,q′ 2 p )for 1 j m, for 1 l 1 , l 2 n, for 1 r 1 , r 2 p and for (z, c) ∈ M .Here, R j l 1 ,k and R∗j r 1 ,q are universal polynomials. Then applying once more

Cramer’s rule, we get(11.13) ⎧∂ 2 Π ′ j⎪⎨(∂ 2 Π ′ ∗j( ) S ∗j r 1 ,r 2⎪⎩ ∂a ′ r 1a ′ r 2 f(c), ϕ(z) = [1k ′ 1 ,k′ 2 n )(( ) S j {Lk′l 1 ,l 2 1L k ′2ϕ i′ (z) } 1i ′ n+m∂x ′ l 1x ′ l 2φ(z), h(c) = [det ( L k ′ φ l′ (z) ) ]1l ′ n 3and1k ′ n{L ∗ qL ∗ 1′ q 2′h i′ (c)} 1i′ p+mdet ( L ∗ q ′ f r′ (c) ) 1r ′ p1q ′ p1q ′ 1 ,q′ 2 p )] 3.By induction, for every j with 1 j m and every two multiindicesβ ∈ N n and δ ∈ N p , there exists two universal polynomials S j β and S∗ jδ suchthat⎧( {L∂ |β| Π ′ j( ) S j β ′βϕ i′ (z) } )1i ′ n+m|β⎪⎨∂x ′ β φ(z), h(c) = ′ ||β|[det ( L k ′ φ l′ (z) ) ]1l ′ n 2|β|+1and1k(11.14)′ n(∂ |γ| Π ′ ∗j( ) S ∗ j {L∗δ ′δh i′ (c) } )1i ′ p+m|δ⎪⎩ ∂a ′ δ f(c), ϕ(z) = ′ ||δ|[det ( L ∗ qf ′ r′ (c) ) ]1r ′ p 2|δ|+1.1q ′ pHere, for β ′ ∈ N n , we denote by L β′ the derivation of order |β ′ | defined by(L 1 ) β′ 1 · · ·(Ln ) β′ n . Similarly, for δ ′ ∈ N p , L ∗δ′ denotes the derivation of order|δ ′ | defined by (L ∗ 1) δ′ 1 · · ·(L∗p ) δ′ p .Next, by the assumption that M ′ is solvable with respect to the parameters,there exist integers j(1), . . .,j(p) with 1 j(q) m and multiindicesβ(1), . . ., β(p) ∈ N n with |β(q)| 1 and max 1qp |β(q)| = κ such thatthe local K-analytic map(11.15) ⎛K p+m ∋ c ′ ↦−→ ⎝ ( Π ′j (0, c ′ ) ) () ⎞1jm ∂ |β(q)| Π ′ j(q), (0, c ′ ) ⎠ ∈ K p+m∂x ′ β(q)1qpis of rank p + m at c ′ = 0. Similarly, by the assumption that M ′ is solvablewith respect to the variables, there exist integers j ∼ (1), . . .,j ∼ (n) with1 j ∼ (l) m and multiindices δ(1), . . ., δ(p) ∈ N n with |δ(q)| 1 andmax 1qp |δ(q)| = κ ∗ such that the local K-analytic map(11.16) ((ΠK n+m ∋ z ′ ↦−→′∗j (0, z ′ ) ) ()1jm ∂ |δ(l)| Π ′ ∗j ∼ (l), ∈ K n+m∂a ′ δ(l)(0, z′ )) 1lnis of rank n + m at z ′ = 0. We then consider from the first line of (11.14)only the (p + m) equations written for (j, 0), (j(q), β(q)) and we solve h(c)299

300by means of the analytic implicit function theorem; also, in the second lineof (11.14), we consider the (n+m) equations written for (j, 0), (j ∼ (l), δ(l))and we solve ϕ(z). We get:(11.17)⎧⎪⎨⎪⎩h(c) = Ĥ⎛⎜⎝φ(z),⎛ϕ(z) = ̂Φ ⎜⎝f(c),S j(1)β(1)( {Lβ ′ ϕ i′ (z) } )1i ′ n+m|β ′ ||β(1)|[ (Lkdet ′ φ l′ (z) ) ]1l ′ n 2|β(1)|+1, . . .1k ′ n( {LS j(p) β ′β(p)ϕ i′ (z) } ) ⎞1i ′ n+m|β. . .,′ ||β(p)| ⎟[ (Lkdet ′ φ l′ (z) ) ]1l ′ n 2|β(p)|+1⎠,1k ′ n( {L∗δ ′ h i′ (c) } )1i ′ p+mS ∗ j ∼ (1)δ(1)|δ ′ ||δ(1)|[det ( L ∗ qf ′ r′ (c) ) ]1r ′ p 2|δ(1)|+1, . . .. . .,S ∗ j ∼ (n)δ(n)[1q ′ p( {L∗δ ′ h i′ (c) } ) ⎞1i ′ p+m|δ ′ ||δ(n)| ⎟] 2|δ(n)|+1⎠ ,det ( L ∗ q ′ f r′ (c) ) 1r ′ p1q ′ pfor (z, c) ∈ M . The maps Ĥ and ̂Φ depend only on Π ′ , Π ′∗ .Lemma 11.18. For every β ′ ∈ N n , there exists a universal polynomial P β ′ inthe jet variables J |β′ |z having K-analytic coefficients in (z, c) which dependsonly on Π, Π ∗ such that, for i ′ = 1, . . .,n + m:( )(11.19) L β′ ϕ i′ (z) ≡ P β ′ z, c, J |β′ |z ϕ i′ (z) .A similar property holds for L ∗δ′ h i′ (c).We deduce that there exist two local K-analytic mappings Φ 0 0 and H0 0 suchthat we can write{ (ϕ(z) = Φ00 z, c, Jκ ∗c(11.20)h(c)) ,h(c) = H0( 0 z, c, Jκz ϕ(z) ) ,for (z, c) ∈ M . Concretely, this means that we have two equivalent pairs offormal identities(ϕ(z) ≡ Φ 0 0 z, a, Π ∗ (a, z), Jc κ∗h(a, Π∗ (a, z)) )ϕ ( x, Π(x, c) ) (≡ Φ 0 0 x, Π(x, c), c, Jκ ∗c(11.21)( h(c))h(c) ≡ H00 x, Π(x, c), c, Jκz ϕ(x, Π(x, c)) )h ( a, Π ∗ (a, z) ) (≡ H00 z, a, Π ∗ (a, z), Jz κ ϕ(z))

in K[a, z] n+m and in K[x, c] p+m . We notice that, whereas ϕ and h are apriori ( only purely formal, by construction, Φ 0 0 and H0 0 are K-analytic near0, 0, Jκ ∗c h(0)) and near ( 0, 0, Jz κϕ(0)) .Next, we introduce the following vector fields with K-analytic coefficientstangent to M :⎧V ⎪⎨ j := ∂∂y + ∑ m∂Π ∗l ∂(a, z) j ∂yj ∂bl, j = 1, . . .,m and(11.22)⎪⎩l=1V ∗ j := ∂∂b j + m∑l=1∂Π l ∂(x, c)∂bj ∂y l,Indeed, we check that V j1 [b j 2− Π ∗j 2(a, z)] ≡Π j 2(x, c)] ≡ 0.j = 1, . . .,m.3010 and that V ∗ j 1[y j 2−For δ ′ ∈ N m , we observe that V δ′ ϕ = ∂|δ′| ϕ. Applying then L β′ with∂y δ′β ′ ∈ N n , we get for i = 1, . . .,n + m:((11.23) L β′ V δ′ ϕ i (z) = Q β ′ ,δ ′ z, c, J|β ′ |+|δ ′ |z ϕ i (z) ) ,with Q β ′ ,δ ′ universal. Since the n + m vector fields L k and V j , having coefficientsdepending on (z, c), span the tangent space to K n x × Km y , the changeof basis of derivations yields, by induction, the following.Lemma 11.24. For every α ∈ N n+m , there exists a universal polynomial P αin its last variables with coefficients being K-analytic in (z, c) and dependingonly on Π, Π ∗ such that, for i = 1, . . ., n + m:(11.25) ∂z α ϕi (z) ≡ P α(z, c, ( L β′ V δ′ ϕ i (z) ) ).|β ′ |+|δ ′ ||α|We are now in position to state and to prove the first fundamental technicallemma which generalizes the two formulas (11.20) to arbitrary jets.Lemma 11.26. For every λ ∈ N, there exist two local K-analytic maps, Φ λ 0valued in K (n+m)Cλ n+m+λ , and Hλ0 valued in K (p+m)Cλ p+m+λ , such that:{ (Jλz ϕ(z) ≡ Φ λ 0 z, c, Jκ ∗ +λc h(c) ) ,(11.27)Jc λ h(c) ≡ ( Hλ 0 z, c, Jκ+λz ϕ(z) ) .Proof. Consider for instance the first line. To obtain it, it suffices to applythe derivations L β′ V δ′ with |β ′ | + |δ ′ | λ to the first line of (11.20), to usethe chain rule and to apply Lemma 11.24.Let θ ∈ K l , l ∈ N, let Q(θ) = ( Q 1 (θ), . . .,Q n+2m+p (θ) ) ∈ K[θ] n+2m+pand let a 1 ∈ K p . As the multiple flow of L ∗ given by (10.10) does not act onthe variables (x, y), we have the trivial but crucial property:(11.28)ϕ ( L ∗ a 1(Q(θ)) ) ≡ ϕ ( π z (L ∗ a 1(Q(θ))) ) ≡ ϕ (π z (Q(θ))) ≡ ϕ (Q(θ)) .

302At the end, we allow to suppress the projection π z : this slight abuse of notationwill lighten slightly the writting of further formulas. More generally,for λ ∈ N, a 1 ∈ K p , x 1 ∈ K n :(11.29)J λ z ϕ( L ∗ a 1(Q(θ)) ) ≡ J λ z ϕ( Q(θ) )J λ c h ( L x1 (Q(θ)) ) ≡ J λ c h ( Q(θ) ) .As a consequence, for 2k even and for 2k + 1 odd, we have the followingfour cancellation relations, useful below (we drop π z and π c after J λ z ϕ andafter J λ c h):and(11.30)⎧Jz λ ϕ( Γ 2k ([xa] 2k ) ) ≡ Jz λ ϕ( Γ 2k−1 ([xa] 2k−1 ) ) ,⎪⎨ Jc λ h ( Γ ∗ 2k([ax] 2k ) ) ≡ Jc λ h ( Γ ∗ 2k−1([ax] 2k−1 ) ) ,Jz λ ⎪⎩ϕ( Γ ∗ 2k+1 ([ax] 2k+1) ) ≡ Jz λ ϕ( Γ ∗ 2k ([ax] 2k) ) ,Jc λ h( Γ 2k+1 ([xa] 2k+1 ) ) ≡ Jc λ h( Γ 2k ([xa] 2k ) ) .We are now in position to state and to prove the second main technicalproposition.Proposition 11.31. For every even chain-length 2k and for every jet-heightλ, there exist two local K-analytic maps, Φ λ 2k valued in K(n+m)Cλ n+m+λ , andH λ 2k valued in K(p+m)Cλ p+m+λ such that:(11.32){Jλz ϕ ( ( )) (Γ ∗ 2k [ax]2k ≡ Φλ2k [ax]2k , J k(κ+κ∗ )+λz ϕ(0) ) andJ λ c h ( Γ 2k([xa]2k))≡ Hλ2k([xa]2k , J k(κ+κ∗ )+λc ϕ(0) ) .Similarly, for every odd chain length 2k + 1 and for every jet eight λ, thereexist two local K-analytic maps, Φ λ 2k+1 valued in K(n+m)Cλ n+m+λ and Hλ2k+1valued in K (p+m)Cλ p+m+λ , such that:(11.33) {Jλz ϕ ( ( )) (Γ 2k+1 [xa]2k+1 ≡ Φλ2k+1 [xa]2k+1 , J kκ+(k+1)κ∗ +λc h(0) ) ,Jc λ h ( ( )) (Γ ∗ 2k+1 [ax]2k+1 ≡ Hλ2k+1 [ax]2k+1 , J (k+1)κ+kκ∗ +λz ϕ(0) ) .These maps depend only on Π, Π ∗ , Π ′ , Π ′∗ .Proof. For 2k + 1 = 1, we replace (z, c) by Γ 1 ([xa] 1 ) in the first lineof (11.27) and by Γ ∗ 1 ([ax] 1) in the second line. Taking crucially account

of the cancellation properties (11.29), we get:⎧Jz λ ϕ( Γ 1 ([xa] 1 ) ) (≡ Φ λ 0 Γ1 ([xa] 1 ), J κ∗ +λc h ( Γ 1 ([xa] 1 ) ))(≡ Φ λ 0 Γ1 ([xa] 1 ), J κ∗ +λc h(0)) )⎪⎨(=: Φ λ 1 [xa]1 , J κ∗ +λc h(0) ) ,(11.34)Jc λ h( Γ ∗ 1 ([ax] 1) ) (≡ H0λ Γ∗1 ([ax] 1 ), Jz κ+λ ϕ ( Γ ∗ 1 ([ax] 1) ))(≡ H0λ Γ∗1 ([ax] 1 ), Jz κ+λ ϕ(0) )⎪⎩(=: H1λ [ax]1 , Jz κ+λ ϕ(0) ) .Here, the third line defines Φ λ 1 and the sixth line defines Hλ 1 . Thus, theproposition holds for 2k + 1 = 1.The rest of the proof proceeds by induction. We treat only the inductionstep from an odd chain-length 2k + 1 to an even chain-length 2k + 2, theother induction step being similar.To this aim, we replace the variables (z, c) in the first line of (11.27)by Γ ∗ 2k+2 ([ax] 2k+2). Taking account of the cancellation property and of theinduction assumption:(11.35)Jz λ ϕ ( Γ ∗ ( ))2k+2 [ax]2k+2 ≡ Φλ0(Γ ∗ 2k+2 ([ax] 2k+2), J κ∗ +λc h ( Γ ∗ 2k+2 ([ax] 2k+2) ))(≡ Φ λ 0 Γ ∗ 2k+2 ([ax] 2k+2), J κ∗ +λc h ( Γ ∗ 2k+1 ([ax] 2k+1) ))(())≡ Φ λ 0 Γ ∗ 2k+2 ([ax] 2k+2), H κ∗ +λ[ax] 2k+1 , J (k+1)(κ+κ∗ )+λc ϕ(0)=: Φ λ 2k+2(2k+1[ax] 2k+2 , J (k+1)(κ+κ∗ )+λc2k+1)ϕ(0) ,The last line defines Φ λ 2k+2 . Similarly, we replace (z, c) in the second lineof (11.27) by Γ 2k+2 ([xa] 2k+2 ). Taking account of the cancellation propertyand of the induction assumption:(11.36)Jc λ h ( Γ 2k+2 ([xa] 2k+2 ) ) (≡ H0λ Γ 2k+2 ([xa] 2k+2 ), Jc κ+λ ϕ ( Γ 2k+2 ([xa] 2k+2 ) ))(≡ H0λ Γ 2k+2 ([xa] 2k+2 ), Jc κ+λ ϕ ( Γ 2k+1 ([xa] 2k+1 ) ))(())≡ H0λ Γ 2k+2 ([xa] 2k+2 ), Φ κ+λ [xa] 2k+1 , J (k+1)(κ+κ∗ )+λc h(0)This completes the proof.=: H λ 2k+2([xa] 2k+2 , J (k+1)(κ+κ∗ )+λc)h(0) .End of the proof of Theorem 11.6. With (µ, µ ∗ ) being the type of (F v , F p )and with [ax] 0 2µ ∗ given by Corollary 10.31, the rank property (10.32) insuresthe existence of an affine (n+m)-dimensional space H ⊂ K µ∗ (p+n) passing303

304through [ax] 0 2µ ∗ and equipped with a local parametrization(11.37) K n+m ∋ s ↦→ [ax] 2µ ∗(s) ∈ Hsatisfying [ax] 2µ ∗(0) = [ax] 0 2µ ∗, such that the map(11.38) K n+m ∋ s ↦−→ π z(Γ∗2µ ∗([ax] 2µ ∗(s)) ) =: z(s) ∈ K n+mis a local diffeomorphism fixing 0 ∈ K n+m . Replacing z by z(s) in ϕ(z)and applying the formula in the first line of (11.32) with λ = 0 and withk = 2µ ∗ , we obtainϕ(z(s)) = ϕ ( (π z Γ∗2µ ∗([ax] 2µ ∗(s) ))(11.39)= ϕ ( (Γ ∗ 2µ ∗ [ax]2µ ∗(s) ))≡ Φ 0 2µ ∗ ([ax]2µ ∗(s), J µ∗ (κ+κ ∗ )z ϕ(0) ) .Inverting s ↦→ z = z(s) as z ↦→ s = s(z), we finally get(ϕ(z) = ϕ(z(s(z))) ≡ Φ 0 2µ(11.40)∗ [ax]2µ ∗(s(z)), J µ∗ (κ+κ ∗ )z ϕ(0) )(=: Φ l z, Jµ ∗ (κ+κ ∗ )z ϕ(0) ) ,with l := µ ∗ (κ + κ ∗ ), where the last line defines Φ l . In conclusion, we havederived the first line of (11.7). The second one is obtained similarly.If Π, Π ∗ , Π ′ , Π ′∗ are algebraic, so are Γ k , Γ ∗ k , Ĥ, ̂Φ, Φ λ 0 , Hλ 0 , Φλ k , Hλ k andΦ l , H l ∗.The proof of Theorem 11.6 is complete.

305II: Explicit prolongations of infinitesimal Lie symmetriesTable of contents1. Jet spaces and prolongations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.2. One independent variable and one dependent variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.3. Several independent variables and one dependent variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320.4. One independent variable and several dependent variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339.5. Several independent variables and several dependent variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348.§1. JET SPACES AND PROLONGATIONS1.1. Choice of notations for the jet space variables. Let K = R or C. Letn 1 and m 1 be two positive integers and consider two sets of variablesx = (x 1 , . . .,x n ) ∈ K n and y = (y 1 , . . .,y m ). In the classical theoryof Lie symmetries of partial differential equations, one considers certaindifferential systems whose (local) solutions should be mappings of the formy = y(x). We refer to [Ol1986] and to [BK1989] for an exposition of thefundamentals of the theory. Accordingly, the variables x are usually calledindependent, whereas the variables y are called dependent. Not to enter insubtle regularity considerations (as in [Me2005b]), we shall assume C ∞ -smoothness of all functions throughout this paper.Let κ 1 be a positive integer. For us, in a very concrete way (withoutfiber bundles), the κ-th jet space Jn,m κ consists of the space Kn+m+m(n+m)! n! m!equipped with the affine coordinates((1.2) x i , y j , y j i 1, y j i 1 ,i 2, . . ....,y j )i 1 ,i 2 ,...,i κ ,having the symmetries(1.3) y j i 1 ,i 2 ,...,i λ= y j i σ(1) ,i σ(2) ,...,i σ(λ),for every λ with 1 λ κ and for every permutation σ of the set{1, . . ., λ}. The variable y j i 1 ,i 2 ,...,i λis an independent coordinate correspondingto the λ-th partial derivativeλ y j. So the symmetries (1.3) are∂∂x i 1∂x i 2···∂x i λnatural.In the classical Lie theory ([OL1979], [Ol1986], [BK1989]), all the geometricobjects: point transformations, vector fields, etc., are local, defined ina neighborhood of some point lying in some affine space K N . However, inthis paper, the original geometric motivations are rapidly forgotten in orderto focus on combinatorial considerations. Thus, to simplify the presentation,we shall not introduce any special notation to speak of certain local

306open subsets of K n+m , or of the jet space Jn,m κ = Kn+m+m(n+m)! n! m! , etc.: wewill always work in global affine spaces K N .1.4. Prolongation ϕ (κ) of a local diffeomorphism ϕ to the κ-th jet space.In this paragraph, we recall how the prolongation of a diffeomorphism to theκ-th jet space is defined ([OL1979], [Ol1986], [BK1989]).Let x ∗ ∈ K n be a central fixed point and let ϕ : K n+m → K n+m bea diffeomorphism whose Jacobian matrix is close to the identity matrix, atleast in a small neighborhood of x ∗ . Let(1.5) Jx κ ∗:= ( x i ∗ , ) ∣ yj ∗i 1, y j ∗i 1 ,i 2, . . ....,y j ∗i 1 ,i 2 ,...,iκ ∈ Jκ n,m x ∗be an arbitrary κ-jet based at x ∗ . The goal is to defined its transformationϕ (κ) (Jx κ ∗) by ϕ.To this aim, choose an arbitrary mapping K n ∋ x ↦→ g(x) ∈ K m definedat least in a neighborhood of x ∗ and representing this κ-th jet, i.e. satisfying(1.6) y j ∂ λ g j∗i 1 ,...,i λ= (x∂x i 1 · · ·∂xi ∗ ),λfor every λ ∈ N with 0 λ κ, for all indices i 1 , . . .,i λ with 1 i 1 , . . .,i λ n and for every j ∈ N with 1 j m. In accordance withthe splitting (x, y) ∈ K n × K m of coordinates, split the components of thediffeomorphism ϕ as ϕ = (φ, ψ) ∈ K n × K m . Write (x, y) the coordinatesin the target space, so that the diffeomorphism ϕ is:(1.7) K n+m ∋ (x, y) ↦−→ (x, y) = ( φ(x, y), ψ(x, y) ) ∈ K n+m .Restrict the variables (x, y) to belong to the graph of g, namely put y := g(x)above, which yields{ x = φ(x, g(x)),(1.8)y = ψ(x, g(x)).As the differential of ϕ at x ∗ is close to the identity, the first family of nscalar equations may be solved with respect to x, by means of the implicitfunction theorem. Denote x = χ(x) the resulting mapping, satisfying bydefinition(1.9) x ≡ φ (χ(x), g(χ(x))).Replace x by χ(x) in the second family of m scalar equations (1.8) above,which yields:(1.10) y = ψ (χ(x), g(χ(x))) .Denote simply by y = g(x) this last relation, where g(·) :=ψ (χ(·), g(χ(·))).In summary, the graph y = g(x) has been transformed to the graph y =g(x) by the diffeomorphism ϕ.

Define then the transformed jet ϕ ( ) (κ) Jx κ ∗ to be the κ-th jet of g at thepoint x ∗ := φ(x ∗ ), namely:(1.11) ϕ ( ()) (κ) Jx κ ∂ λ g j 1jm∗:=∂x i1 · · ·∂x i (x ∗)∈ J κ ∣λ n,m x∗.1i 1 ,...,i λ n, 0λκIt may be shown that this jet does not depend on the choice of a localgraph y = g(x) representing the κ-th jet Jx κ ∗at x ∗ . Furthermore, ifπ κ := Jn,m κ → Km denotes the canonical projection onto the first factor,the following diagram commutes:Jn,mκ ϕ (κ) Jn,mκπ κ π κ.K n+m ϕ K n+m1.12. Inductive formulas for the κ-th prolongation ϕ (κ) . To present them,we change our notations. Instead of (x, y), as coordinates in the target spaceK n × K m , we shall use capital letters:((1.13)X 1 , . . .,X n , Y 1 , . . .,Y m) .In the source space K n+m equipped with the coordinates (x, y), we use thejet coordinates (1.2) on the associated κ-th jet space. In the target spaceK n+m equipped with the coordinates (X, Y ), we use the coordinates((1.14) X i , Y j , Y j , Y j X i 1 X i 1X , . . ....,Y )ji 2 X i 1X i 2...X iκon the associated κ-th jet space; to avoid confusion with y i1 , y i1 ,i 2, . . . insubsequent formulas, we do not write Y i1 , Y i1 ,i 2, . . . . In these notations, thediffeomorphism ϕ whose first order approximation is close to the identitymapping in a neighborhood of x ∗ may be written under the form:(1.15) ϕ : ( ) (x i′ , y j′ ↦→ X i , Y j) ()= X i (x i′ , y j′ ), Y j (x i′ , y j′ ) ,for some C ∞ -smooth functions X i (x i′ , y j′ ), i = 1, . . .,n, and Y j (x i′ , y j′ ),j = 1, . . ., m. The first prolongation ϕ (1) of ϕ may be written under theform:(1.16) ( ) (( ))ϕ (1) : x i′ , y j′ , y j′i↦−→ X i (x i′ , y j′ ), Y j (x i′ , y j′ ), Y j x i′ , y j′ , y j′′1X i 1 i, ′1( )for some functions Y j x i′ , y j′ , y j′X i 1 iwhich depend on the pure first jet′1variables y j′iThe way how these functions depend on the first order partialderivatives functions X i , x i′ Xi , Y j , Y j and on the pure first jet1. ′ y j′ x i′ y j′307

308variables y j′iis provided (in principle) by the following compact formulas′1([BK1989]):⎛ ⎛(1.17)⎜⎝⎞Y j XD1X 1 1 · · · D1X 1 n ⎞1⎟. ⎠ = ⎝. · · · .⎠Y j D 1 X n n X1 · · · Dn 1Xn −1 ⎛⎝D1Y 1 j ⎞.⎠ ,Dn 1Y jwhere, for i ′ = 1, . . ., n, the symbol Di 1 denotes the ′ i′ -th first order totaldifferentiation operator:(1.18) Di 1 := ∂ m∑ ∂+ y j′′∂x i′ i.′∂y j′Striclty speaking, these formulas (1.17) are not explicit, because an inversematrix is involved and because the terms Di 1 ′Xi , Di 1 ′Y j are not developed.However, it would be feasible and elementary to write down thecorresponding totally explicit complete formulas for the functions Y j = X i 1Y j X i 1().j ′ =1x i′ , y j′ , y j′i ′ 1Next, the second prolongation ϕ (2) is of the form(1.19)ϕ (2) :)(x i′ , y j′ , y j′i, y j′′1 i ′ 1 ,i′ 2for some functions Y j X i 1X i 2↦−→()(ϕ(x (1) i′ , y j′ , y j′i ′ 1)x i′ , y j′ , y j′i ′ 1, y j′i ′ 1 ,i′ 2, Y j X i 1X i 2))(x i′ , y j′ , y j′i, y j′′1 i,′1 ,i′ 2which depend on the purefirst and second jet variables. For i = 1, . . .,n, the expressions of Y j X i 1Xare given by the following compact formulas (again [BK1989]):i⎛ ⎞Y j ⎛DX⎜i 1X 11 1⎟X1 · · · D 1 ⎞ ⎛ ⎞1 Xn−1D1Y 2 j X(1.20) ⎝ . ⎠ = ⎝. · · · .⎠ ⎜ i 1⎟ ⎝ . ⎠ ,Y j D 1 X i 1X nn X1 · · · Dn 1Xn DnY 2 j X i 1where, for i ′ = 1, . . .,n, the symbol Di 2 denotes the ′ i′ -th second order totaldifferentiation operator:(1.21) Di 2 := ∂ m∑ ∂m∑ n∑ ∂+ y j′′∂x i′ i+.′∂y j′j ′ =1j ′ =1i ′ 1 =1 y j′i ′ ,i ′ 1∂y j′i ′ 1Again, these formulas (1.20) are not explicit in the sense that an inversematrix is involved and that the terms Di 1 ′Xi , Di 2 ′Y j are not developed. ItX i 1would already be a nontrivial computational task to develope these expressionsand to find out some nice satisfying combinatorial formulas.In order to present the general inductive non-explicit formulas for thecomputation of the κ-th prolongation ϕ (κ) , we need some more notation.

Let λ ∈ N be an arbitrary integer. For i ′ = 1, . . ., n, let D λ i ′ denotes the i′ -thλ-th order total differentiation operators, defined precisely by:(1.22) ⎧⎪⎨⎪⎩D λ i ′ := ∂∂x i′ ++ · · · +m∑j ′ =1m∑j ′ =1y j′i ′∂∂y j′ +n∑m∑j ′ =1n∑i ′ 1 =1 y j′i ′ ,i ′ 1i ′ 1 ,i′ 2 ,...,i′ λ−1 =1 y j′i ′ ,i ′ 1 ,i′ 2 ,...,i′ λ−1∂∂y j′i ′ 1+m∑j ′ =1∂∂y j′i ′ 1 ,i′ 2 ,...,i′ λ−1n∑i ′ 1 ,i′ 2 =1 y j′i ′ ,i ′ 1 ,i′ 2.309∂∂y j′i ′ 1 ,i′ 2Then, for i = 1, . . .,n, the expressions of Y j are given by theX i 1 ···X i λ−1Xi following compact formulas (again [BK1989]):(1.23) ⎛⎛⎞⎜⎝Y j ⎞DX i 1 ···X i λ−1X 11 1⎟X1 · · · D 1 ⎞ ⎛1 Xn−1D1 λ. ⎠ = ⎝. · · · .⎠ ⎜ Y j X i 1 ···X i λ−1⎝ .Y j D 1X i 1 ···X i λ−1XnX 1 · · · DnX 1 n D λ n n Y j X i 1 ···X i λ−1Again, these inductive formulas are incomplete and unsatisfactory.Problem 1.24. Find totally explicit complete formulas for the κ-th prolongationϕ (κ) .⎟⎠ .Except in the cases κ = 1, 2, we have not been able to solve this problem.The case κ = 1 is elementary. Complete formulas in the particularcases κ = 2, n = 1, m 1 and n 1, m = 1 are implicitely providedin [Me2004] and in Section ?(?), where one observes the appearance of somemodifications of the Jacobian determinant of the diffeomorphism ϕ, insertedin a clearly understandable combinatorics. In fact, there is a nice dictionarybetween the formulas for ϕ (2) and the formulas for the second prolongationL (2) of a vector field L which were written in equation (43) of [GM2003a](see also equations (2.6), (3.20), (4.6) and (5.3) in the next paragraphs). Inthe passage from ϕ (2) to L (2) , a sort of formal first order linearization maybe observed and the reverse passage may be easily guessed. However, forκ 3, the formulas for ϕ (κ) explode faster than the formulas for the κ-thprolongation L (κ) of a vector field L . Also, the dictionary between ϕ (κ)and L (κ) disappears. In fact, to elaborate an appropriate dictionary, webelieve that one should introduce before a sort of formal (κ − 1)-th orderlinearizations of ϕ (κ) , finer than the first order linearization L (κ) . To be optimistic,we believe that the final answer to Problem 1.24 is, nevertheless,accessible after hard work.The present article is devoted to present totally explicit complete formulasfor the κ-th prolongation L (κ) of a vector field L to J κ n,m , for n 1arbitrary, for m 1 arbitrary and for κ 1 arbitrary.+

3101.25. Prolongation of a vector field to the κ-th jet space. Consider a vectorfield(1.26) L =n∑i=1X i (x, y) ∂∂x i + m∑j=1Y j (x, y) ∂∂y j ,defined in K n+m . Its flow:(1.27) ϕ t (x, y) := exp (tL )(x, y)constitutes a one-parameter family of diffeomorphisms of K n+m close tothe identity. The lift (ϕ t ) (κ) to the κ-th jet space constitutes a one-parameterfamily of diffeomorphisms of J κ n,m. By definition, the κ-th prolongationL (κ) of L to the jet space J κ n,m is the infinitesimal generator of (ϕ t) (κ) ,namely:(1.28) L (κ) := d [ dt∣ (ϕt ) (κ)] .t=01.29. Inductive formulas for the κ-th prolongation L (κ) . As a vector fielddefined in K n+m+m(n+m)! n! m! , the κ-th prolongation L (κ) may be written underthe general form:(1.30) ⎧ n∑L (κ) = X i ∂∂x + ∑ mY j ∂i ∂y + j⎪⎨⎪⎩i=1++m∑j=1m∑j=1n∑i 1 =1n∑Y j i 1i 1 ,...,i κ=1j=1∂∂y j i 1+Y j i 1 ,...,i κm∑j=1n∑i 1 ,i 2 =1∂∂y j i 1 ,...,i κ.Y j i 1 ,i 2∂∂y j i 1 ,i 2+ · · ·+Here, the coefficients Y j i 1, Y j i 1 ,i 2, . . . , Y j i 1 ,i 2 ,...,i κare uniquely determinedin terms of partial derivatives of the coefficients X i and Y j of the originalvector field L , together with the pure jet variables ( y j i 1, . . .,y j i 1 ,...,i κ),by means of the following fundamental inductive formulas ([OL1979],

311[Ol1986], [BK1989]):(Y(1.31)⎧⎪ j i 1:= D ) n∑ (i 1 1 Yj− D ) i 1 1 Xky j k ,k=1⎨( )n∑ (Y j i 1 ,i 2:= Di 2 2 Yji 1 − D ) i 1 2 Xky j i 1 ,k ,k=1· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·( ) n∑⎪ ⎩ Y j i 1 ,i 2 ,...,i κ:= Di κ κY j (i 1 ,i 2 ,...,i κ−1− D ) i 1 κ Xky j i 1 ,i 2 ,...,i κ−1 ,k ,where, for every λ ∈ N with 0 λ κ, and for every i ∈ N with 1 i ′ n, the i ′ -th λ-th order total differentiation operator Di λ ′ was defined in (1.22)above.Problem 1.32. Applying these inductive formulas, find totally explicit completeformulas for the κ-th prolongation L (κ) .The present article is devoted to provide all the desired formulas.1.33. Methodology of induction. We have the intention of presenting ourresults in a purely inductive style, based on several thorough visual comparisonsbetween massive formulas which will be written and commented infour different cases:k=1(i) n = 1 and m = 1; κ 1 arbitrary;(ii) n 1 and m = 1; κ 1 arbitrary;(iii) n = 1 and m 1; κ 1 arbitrary;(iv) general case: n 1 and m 1; κ 1 arbitrary.Accordingly, we shall particularize and slightly lighten our notations ineach of the three (preliminary) cases (i) [Section 2], (ii) [Section 3] and (iii)[Section 4].§2. ONE INDEPENDENT VARIABLE AND ONE DEPENDENT VARIABLE2.1. Simplified adapted notations. Assume n = 1 and m = 1, let κ ∈ Nwith κ 1 and simply denote the jet variables by:(2.2) (x, y, y 1 , y 2 , . . .,y κ ) ∈ J κ1,1 .The κ-th prolongation of a vector field L = X ∂ + Y ∂ will be denoted∂x ∂yby:(2.3) L (κ) = X ∂∂x + Y ∂∂y + Y 1∂∂y 1+ Y 2∂∂y 2+ · · · + Y κ∂∂y κ.

312The coefficients Y 1 , Y 2 , . . . , Y κ are computed by means of the inductiveformulas:⎧Y 1 := D 1 (Y ) − D 1 (X ) y 1 ,⎪⎨Y 2 := D 2 (Y 1 ) − D 1 (X ) y 2 ,(2.4)· · · · · · · · · · · · · · · · · · · · · · · · · · ·⎪⎩Y κ := D κ (Y κ−1 ) − D 1 (X ) y κ ,where, for 1 λ κ:(2.5) D λ := ∂∂x + y 1∂∂y + y 2∂∂y 1+ · · · + y λ∂∂y λ−1.By direct elementary computations, for κ = 1 and for κ = 2, we obtain thefollowing two very classical formulas :(2.6) ⎧⎪⎨Y 1 = Y x + [Y y − X x ] y 1 + [−X y ] (y 1 ) 2 ,Y 2 = Y x 2 + [2 Y xy − X x 2] y 1 + [Y y 2 − 2 X xy ] (y 1 ) 2 + [−X y 2] (y 1 ) 3 +⎪⎩+ [Y y − 2 X x ] y 2 + [−3 X y ] y 1 y 2 .Our main objective is to devise the general combinatorics. Thus, to attainthis aim, we have to achieve patiently formal computations of the next coefficientsY 3 , Y 4 and Y 5 . We systematically use parentheses [·] to single outevery coefficient of the polynomials Y 3 , Y 4 and Y 5 in the pure jet variablesy 1 , y 2 , y 3 , y 4 and y 5 , putting every sign inside these parentheses. We alwaysput the monomials in the pure jet variables y 1 , y 2 , y 3 , y 4 and y 5 after theparentheses. For completeness, let us provide the intermediate computationof the third coefficient Y 3 . In detail:Y 3 = D 3 (Y 2 ) − D 1 (X ) y 3( ∂=∂x + y ∂1∂y + y 2∂∂y 1+ y 3+ [Y y 2 − 2 X xy ] (y 1 ) 2 + [−X y 2](y 1 ) 3 ++ [Y y − 2 X x ]y 2 + [−3 X y ] y 1 y 2)∂∂y 2) (Y x 2 + [2 Y xy − X x 2] y 1 +(2.7)= Y x 3 1+ [2 Y x 2 y − X x 3] y 1 2+ [Y xy 2 − 2 X x 2 y] (y 1 ) 2 3 + [−X xy 2] (y 1) 3 4 ++ [Y xy − 2 X x 2]y 2 6+ [−3 X xy ] y 1 y 2 7+ [Y x 2 y]y 1 2++ [2 Y xy 2 − X x 2 y] (y 1 ) 2 3 + [Y y 3 − 2 X xy 2] (y 1) 3 4 + [−X y 3] (y 1) 4 5 +

+ [Y y 2 − 2 X xy ] y 1 y 2 + [−3 X y 2](y 1 ) 2 y 2 + [2 Y xy − X x 2] y 2 +7 8 6+ [Y y 2 − 2 X xy ] 2 y 1 y 2 + [−X y 2]3(y 1 ) 2 y 2 + [−3 X y ] (y 2 ) 2 +7 8 9+ [Y y − 2 X x ] y 3 + [−3 X y ] y 1 y 3 −10 11− [X x ] y 3 10− [X y ]y 1 y 3 . 11313We have underlined all the terms with a number appended. Each numberrefers to the order of appearance of the terms in the final simplified expressionof Y 3 , also written in [BK1989] with different notations:(2.8) ⎧Y 3 = Y x 3 + [3 Y x⎪⎨2 y − X x 3] y 1 + [3 Y xy 2 − 3 X x 2 y] (y 1 ) 2 ++ [Y y 3 − 3 X xy 2](y 1 ) 3 + [−X y 3](y 1 ) 4 + [3 Y xy − 3 X x 2]y 2 +⎪⎩+ [3 Y y 2 − 9 X xy ] y 1 y 2 + [−6 X y 2] (y 1 ) 2 y 2 + [−3 X y ] (y 2 ) 2 ++ [Y y − 3 X x ] y 3 + [−4 X y ]y 1 y 3 .After similar manual computations, the intermediate details of which wewill not copy in this Latex file, we get the desired expressions of Y 4 and ofY 5 . Firstly:(2.9)⎧Y 4 = Y x 4 + [ 4Y x 3 y − X x 4]y1 + [ 6Y x 2 y 2 − 4X x y] 3 (y1 ) 2 +⎪⎨⎪⎩+ [ 4Y xy 3 − 6X x 2 y 2 ](y1 ) 3 + [ Y y 4 − 4X xy 3](y1 ) 4 + [ −X y 4](y1 ) 5 ++ [ 6Y x 2 y − 4X x 3]y2 + [ 12Y xy 2 − 18X x 2 y]y1 y 2 ++ [ 6Y y 3 − 24X xy 2](y1 ) 2 y 2 + [ −10X y 3](y1 ) 3 y 2 ++ [ 3Y y 2 − 12X xy](y2 ) 2 + [ −15X y 2]y1 (y 2 ) 2 ++ [4Y xy − 6X x 2] y 3 + [ 4Y y 2 − 16X xy]y1 y 3 + [ −10X y 2](y1 ) 2 y 3 ++ [−10X y ]y 2 y 3 + [Y y − 4X x ] y 4 + [−5X y ] y 1 y 4 .

314Secondly:(2.10)⎧Y 5 = Y x 5 + [ 5Y x 4 y − X x 5]y1 + [ 10Y x 3 y 2 − 5X x y] 4 (y1 ) 2 +⎪⎨⎪⎩+ [ 10Y x 2 y 3 − 10X x 3 y 2 ](y1 ) 3 + [ 5Y xy 4 − 10X x 2 y 3 ](y1 ) 4 ++ [ Y y 5 − 5X xy 4](y1 ) 5 + [ −X y 5](y1 ) 6 + [ 10Y x 3 y − 5X x 4]y2 ++ [ 30Y x 2 y 2 − 30X x 3 y]y1 y 2 + [ 30Y xy 3 − 60X x 2 y 2 ](y1 ) 2 y 2 ++ [ 10Y y 4 − 50X xy 3](y1 ) 3 y 2 + [ −15X y 4](y1 ) 4 y 2 ++ [ 15Y xy 2 − 30X x 2 y](y2 ) 2 + [ 15Y y 3 − 75X xy 2]y1 (y 2 ) 2 ++ [ −45X y 3](y1 ) 2 (y 2 ) 2 + [ −15X y 2](y2 ) 3 ++ [ 10Y x 2 y − 10X x 3]y3 + [ 20Y xy 2 − 40X x 2 y]y1 y 3 ++ [ 10Y y 3 − 50X xy 2](y1 ) 2 y 3 + [ −20X y 3](y1 ) 3 y 3 ++ [ 10Y y 2 − 50X xy]y2 y 3 + [ −60X y 2]y1 y 2 y 3 + [−10X y ] (y 3 ) 2 ++ [5Y xy − 10X x 2]y 4 + [ 5Y y 2 − 25X xy]y1 y 4 + [ −15X y 2](y1 ) 2 y 4 ++ [−15X y ]y 2 y 4 + [Y y − 5X y ]y 5 + [−6X y ]y 1 y 5 .2.11. Formal inspection, formal intuition and formal induction. Now,we have to comment these formulas. We have written in length the fivepolynomials Y 1 , Y 2 , Y 3 , Y 4 and Y 5 in the pure jet variables y 1 , y 2 , y 3 , y 4and y 5 . Except the first “constant” term Y x κ, all the monomials in the expressionof Y κ are of the general form(2.12) (y λ1 ) µ 1(y λ2 ) µ2 · · ·(y λd ) µ d,for some positive integer d 1, for some collection of strictly increasing jetindices:(2.13) 1 λ 1 < λ 2 < · · · < λ d κ,and for some positive integers µ 1 , . . .,µ d 1. This and the next combinatorialfacts may be confirmed by reading the formulas giving Y 1 , Y 2 , Y 3 ,Y 4 and Y 5 . It follows that the integer d satisfies the inequality d κ + 1.To include the first “constant” term Y x κ, we shall make the convention thatputting d = 0 in the monomial (2.12) yields the constant term 1.Furthermore, by inspecting the formulas giving Y 1 , Y 2 , Y 3 , Y 4 and Y 5 ,we see that the following inequality should be satisfied:(2.14) µ 1 λ 1 + µ 2 λ 2 + · · · + µ d λ d κ + 1.For instance, in the expression of Y 4 , the two monomials (y 1 ) 3 y 2 and y 1 (y 2 ) 2do appear, but the two monomials (y 1 ) 4 y 2 and (y 1 ) 2 (y 2 ) 2 cannot appear. Allcoefficients of the pure jet monomials are of the general form:[(2.15)A Yx α y − B X ] β+1 x α+1 y , β

for some nonnegative integers A, B, α, β ∈ N. Sometimes A is zero, butB is zero only for the (constant, with respect to pure jet variables) termY x κ. Importantly, X is differentiated once more with respect to x and Y isdifferentiated once more with respect to y. Again, this may be confirmed byreading all the terms in the formulas for Y 1 , Y 2 , Y 3 , Y 4 and Y 5 .In addition, we claim that there is a link between the couple (α, β) andthe collection {µ 1 , λ 1 , . . .,µ d , λ d }. To discover it, let us write some of themonomials appearing in the expressions of Y 4 (first column) and of Y 5(second column), for instance:(2.16)⎧[6 Y x 2 y 2 − 4 X x 3 y](y 1 ) 2 , [5 Y xy 4 − 10 X x 2 y 3] (y 1) 4 ,⎪⎨[12 Y xy 2 − 18 X x 2 y]y 1 y 2 , [30 Y xy 3 − 60 X x 2 y 2] (y 1) 2 y 2 ,[−10 X y 3] (y 1 ) 3 y 2 , [−15 X y 4] (y 1 ) 4 y 2 ,[4 Y y 2 − 16 X xy ] y 1 y 3 , [10 Y y 2 − 50 X xy ] y 2 y 3 ,⎪⎩[−10 X y 2] (y 1 ) 2 y 3 , [−60 X y 2] y 1 y 2 y 3 .After some reflection, we discover the hidden intuitive rule: the partialderivatives of Y and of X associated with the monomial (y λ1 ) µ1 · · ·(y λd ) µ dare, respectively:{Yx κ−µ 1 λ 1 −···−µ d λ d y µ 1 +···+µ d ,(2.17)X x κ−µ 1 λ 1 −···−µ d λ d +1 y µ 1 +···+µ d −1.This may be checked on each of the 10 examples (2.16) above.Now that we have explored and discovered the combinatorics of the purejet monomials, of the partial derivatives and of the complete sum giving Y κ ,we may express that it is of the following general form:(2.18)⎧Y κ = Y x κ +⎪⎨⎪⎩∑κ+1d=1[A(µ 1 ,λ 1 ),...,(µ d ,λ d )κ∑1λ 1

316(y 1 ) 5 are simple. Indeed, by inspecting the first terms in the expressions ofY 1 , Y 2 , Y 3 , Y 4 and Y 5 , we of course recognize the binomial coefficients.In general:Lemma 2.19. For κ 1,(2.20) ⎧κ∑[( ( ) ]⎪⎨κ κY κ = Y x κ + Yλ)x κ−λ y − X λλ − 1 x κ−λ+1 y (y λ−1 1 ) λ +λ=1⎪⎩+ [−X y κ] (y 1 ) κ + remainder,where the term remainder collects all remaining monomials in the pure jetvariables.In addition, let us remind what we have observed and used in a previousco-signed work.Lemma 2.21. ([GM2003a], p. 536) For κ 4, nine among the monomialsof Y κ are of the following general form:(2.22) ⎧Y κ = Y x κ + [ Cκ 1 Y x κ−1 y − X x κ]y1 + [ Cκ 2 Y x κ−2 y − Cκ 1 X ]x κ−1 y2 +⎪⎨ + [ Cκ 2 Y x 2 y − Cκ 3 X x 3]yκ−2 + [ Cκ 1 Y xy − Cκ 2 X x 2]yκ−1 +⎪⎩+ [ C 1 κ Y y 2 − κ 2 X xy]y1 y κ−1 + [ −C 2 κ X y]y2 y κ−1 ++ [ Y y − C 1 κ X x]+[−C1κ+1 X y]y1 y κ + remainder,where the term remainder denotes all the remaining monomials, and whereCκ λ := κ!is a notation for the binomial coefficient which occupies less(κ−λ)! λ!space in Latex “equation mode” than the classical notation( κ(2.23).λ)Now, we state directly the final theorem, without further inductive or intuitiveinformation.Theorem 2.24. For κ 1, we have:(2.25)∑κ+1∑Y κ = Y x κ +d=11λ 1

Once the correct theorem is formulated, its proof follows by accessibleinduction arguments which will not be developed here. It is better to continuethrough and to examine thorougly the case of several variables, sinceit will help us considerably to explain how we discovered the exact valuesof the integer coefficients A (µ 1,λ 1 ),...,(µ d ,λ d )κ and B (µ 1,λ 1 ),...,(µ d ,λ d )κ .3172.26. Verification and application. Before proceeding further, let usrapidly verify that the above general formula (2.25) is correct by inspectingtwo instances extracted from Y 5 .Firstly, the coefficient of (y 1 ) 3 y 3 in Y 5 is obtained by putting κ = 5,d = 2, λ 1 = 1, µ 1 = 3, λ 2 = 3 and µ 2 = 1 in the general formula (2.25),which yields:(2.27)[0 − 5 · 4 · 3 · 2 · 1 · 6 ](1!) 3 3! (3!) 1 1! X y 3 = [−20 X y 3].This value is the same as in the original formula (2.10): confirmation.Secondly, the coefficient of y 1 (y 2 ) 2 in Y 5 is obtained by κ = 5, d = 2,λ 1 = 1, µ 1 = 1, λ 2 = 2 and µ 2 = 2 in the general formula (2.25), whichyields:(2.28) [ 5 · 4 · 3 · 2 · 1(1!) 1 1! (2!) 2 2! Y y 3 − 5 · 4 · 3 · 2 · 5 ](1!) 1 1! (2!) 2 2! X xy 2 = [15 Y y 3 − 75 X xy 2].This value is the same as in the original formula (2.10); again: confirmation.Finally, applying our general formula (2.25), we deduce the value of Y 6without having to use Y 5 and the induction formulas (2.4), which shortenssubstantially the computations.

318For the pleasure, we obtain:(2.29)⎧Y 6 = Y x 6 + [ 6Y x 5 y − X x 6]y1 + [ 15Y x 4 y 2 − 6X x y] 5 (y1 ) 2 ++ [ 20Y x 3 y 3 − 15X ]x 4 y 2 (y1 ) 3 + [ 15Y x 2 y 4 − 20X ]x 3 y 3 (y1 ) 4 ++ [ ]6Y xy 5 − 15X x 2 y 4 (y1 ) 5 + [ Y y 6 − 6X xy 5](y1 ) 6 + [ −X y 6](y1 ) 7 ++ [ 15Y x 4 y − 6X x 5]y2 + [ 60Y x 3 y 2 − 45X x y] 4 y1 y 2 ++ [ 90Y x 2 y 3 − 120X ]x 3 y 2 (y1 ) 2 y 2 + [ ]60Y xy 4 − 150X x 2 y 3 (y1 ) 3 y 2 ++ [ 15Y y 5 − 90X xy 4](y1 ) 4 y 2 + [ −21X y 5](y1 ) 5 y 2 ++ [ 45Y x 2 y 2 − 60X x y] 3 (y2 ) 2 + [ ]90Y xy 3 − 225X x 2 y 2 y1 (y 2 ) 2 ++ [ 45Y y 4 − 270X xy 3](y1 ) 2 (y 2 ) 2 + [ −210X y 4](y1 ) 3 (y 2 ) 2 ++ [ 15Y y 3 − 90X xy 2](y2 ) 3 + [ −105X y 3]y1 (y 2 ) 3 +⎪⎨ + [ 20Y x 3 y − 15X x 4]y3 + [ 60Y x 2 y 2 − 80X x y] 3 y1 y 3 ++ [ ]60Y xy 3 − 150X x 2 y 2 (y1 ) 2 y 3 + [ 20Y y 4 − 120X xy 3](y1 ) 3 y 3 ++ [ −35X y 4](y1 ) 4 y 3 + [ 60Y xy 2 − 150X x y] 2 y2 y 3 ++ [ 60Y y 3 − 360X xy 2]y1 y 2 y 3 + [ −210X y 3](y1 ) 2 y 2 y 3 ++ [ −105X y 2](y2 ) 2 y 3 + [ ]10Y y 2 − 60X xy (y3 ) 2 ++ [ −70X y 2]y1 (y 3 ) 2 + [ 15Y x 2 y − 20X x 3]y4 ++ [ 30Y xy 2 − 75X x y] 2 y1 y 4 + [ 15Y y 3 − 90X xy 2](y1 ) 2 y 4 ++ [ −35X y 3](y1 ) 3 y 4 + [ ]15Y y 2 − 90X xy y2 y 4 ++ [ −105X y 2]y1 y 2 y 4 + [−35X y ]y 3 y 4 + [6Y xy − 15X x 2]y 5 ++ [ ]6Y y 2 − 36X xy y1 y 5 + [ −21X y 2](y1 )⎪⎩2 y 5 + [−21X y ]y 2 y 5 ++ [Y y − 6X y ]y 6 + [−7X y ] y 1 y 6 .2.30. Deduction of the classical Faà di Bruno formula. Let x, y ∈ K andlet g = g(x), f = f(y) be two C ∞ -smooth functions K → K. Considerthe composition h := f ◦ g, namely h(x) = f(g(x)). For λ ∈ N withλ 1, simply denote by g λ the λ-th derivative dλ gdx λ and similarly for h λ .Also, abbreviate f λ := dλ fdy λ .

By the classical formula for the derivative of a composite function, wehave h 1 = f 1 g 1 . Further computations provide the following list of subsequentderivatives of h:(2.31) ⎧h 1 = f 1 g 1 ,⎪⎨h 4 = f 4 (g 1 ) 4 + 6 f 3 (g 1 ) 2 g 2 + 3 f 2 (g 2 ) 2 + 4 f 2 g 1 g 3 + f 1 g 4 ,h 5 = f 5 (g 1 ) 5 + 10 f 4 (g 1 ) 3 g 2 + 15 f 3 (g 1 ) 2 g 3 + 10 f 3 g 1 (g 2 ) 2 +⎪⎩h 2 = f 2 (g 1 ) 2 + f 1 g 2 ,h 3 = f 3 (g 1 ) 3 + 3 f 2 g 1 g 2 + f 1 g 3 ,+ 10 f 2 g 2 g 3 + 5 f 2 g 1 g 4 + f 1 g 5 ,h 6 = f 6 (g 1 ) 6 + 15 f 5 (g 1 ) 4 g 2 + 45 f 4 (g 1 ) 2 (g 2 ) 2 + 15 f 3 (g 2 ) 3 ++ 20 f 4 (g 1 ) 3 g 3 + 60 f 3 g 1 g 2 g 3 + 10 f 2 (g 3 ) 2 + 15 f 3 (g 1 ) 2 g 4 ++ 15 f 2 g 2 g 4 + 6 f 2 g 1 g 5 + f 1 g 6 .Theorem 2.32. For every integer κ 1, the κ-th derivative of the compositefunction h = f ◦ g may be expressed as an explicit polynomial in the partialderivatives of f and of g having integer coefficients:(2.33)d κ hκ∑dx = ∑ ∑ ∑κd=11λ 1

320Define the differential operators(2.35) ( )∂ ∂F 2 := g 2 + g 1 f 2 ,∂g 1 ∂f 1(∂ ∂ ∂F 3 := g 2 + g 3 + g 1 f 2 + f 3∂g 1 ∂g 2 ∂f 1)∂,∂f 2· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·(∂ ∂ ∂F λ := g 2 + g 3 + · · · + g λ + g 1 f 2∂g 1 ∂g 2 ∂g λ−1Then we have∂∂f 1+ f 3∂∂f 2+ · · · + f λ)∂.∂f λ−1(2.36)h 2 = F 2 (h 1 ),h 3 = F 3 (h 2 ),· · · · · · · · · · · · · · ·h λ = F λ (h λ−1 ).§3. SEVERAL INDEPENDENT VARIABLES AND ONE DEPENDENTVARIABLE3.1. Simplified adapted notations. As announced after the statement ofTheorem 2.24, it is only after we have treated the case of several independentvariables that we will understand perfectly the general formula (2.25), validin the case of one independent variable and one dependent variable. We willdiscover massive formal computations, exciting our computational intuition.Thus, assume n 1 and m = 1, let κ ∈ N with κ 1 and simply denote(instead of (1.2)) the jet variables by:(3.2)(x i , y, y i1 , y i1 ,i 2, . . .,y i1 ,i 2 ,...,i κ).Also, instead of (1.30), denote the κ-th prolongation of a vector field by:(3.3) ⎧L ⎪⎨(κ) =⎪⎩n∑i=1X i+ · · · +∂∂x + Y ∂ n∑i ∂y +n∑i 1 ,i 2 ,...,i κ=1i 1 =1Y i1 ,i 2 ,...,i κY i1∂∂y i1+∂∂y i1 ,i 2 ,...,i κ.n∑i 1 ,i 2 =1Y i1 ,i 2∂∂y i1 ,i 2+

The induction formulas are⎧(3.4)⎪⎨Y i1 := D 1 i 1(Y ) −Y i1 ,i 2:= D 2 i 2(Y i1 ) −n∑ (D ) i 1 1 Xky k ,k=1n∑ (D ) i 1 2 Xky i1 ,k,k=1· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·( )n∑ (⎪⎩ Y i1 ,i 2 ,...,i κ:= Di κ Yi1 κ ,i 2 ,...,i κ−1 − D ) i 1 κ Xky i1 ,i 2 ,...,i κ−1 ,k,are defined as in (1.22), drop-where the total differentiation operators Di λ ′ping the sums ∑ mj ′ =1 and the indices j′ .k=13213.5. Two instructing explicit computations. To begin with, let us computeY i1 . With Di 1 1= ∂ + y ∂∂x i 1 i 1, we have:∂yn∑ (Y i1 = D i1 (Y ) − Di 1 1 Xk 1)yk1(3.6)k 1 =1= Y x i 1 + Y y y i1 −n∑k 1 =1X k 1x i 1 y k 1−n∑k 1 =1X k 1y y i1 y k1 .Searching for formal harmony and for coherence with the formula (2.6) 1 ,we must include the term Y y y i1 inside the sum ∑ nk 1 =1 [·]y k 1. Using theKronecker symbol, we may write:n∑ [(3.7) Y y y i1 ≡ δk 1]i 1Y y yk1 .k 1 =1k 1 =1Also, we may rewrite the last term of (3.6) with a double sum:n∑n∑ [(3.8) − X k 1y y i1 y k1 ≡ −δk 1]i 1X k 2y yk1 y k2 .k 1 ,k 2 =1From now on and up to equation (3.39), we shall abbreviate any sum ∑ nk=1from 1 to n as ∑ k. Putting everything together, we get the final desiredperfect expression of Y i1 :(3.9) Y i1 = Y x i 1 + ∑ [δk 1]i 1Y y − X k 1x i yk1 + ∑ [1−δk 1]i 1X k 2y yk1 y k2 .k 1 k 1 ,k 2This completes the first explicit computation.The second one is about Y i1 ,i 2. It becomes more delicate, because severalalgebraic transformations must be achieved until the final satisfying formulais obtained. Our goal is to present each step very carefully, explaining every

322tiny detail. Without such a care, it would be impossible to claim that some ofour subsequent computations, for which we will not provide the intermediatesteps, may be redone and verified. Consequently, we will expose our rulesof formal computation thoroughly.Replacing the value of Y 1 just obtained in the induction formula (3.4) 2and developing, we may conduct the very first steps of the computation:Y i1 ,i 2= Di 2 2(Y i1 ) − ∑ )Di 1 2(X k 1y i1 ,k 1k 1=⎛⎝ ∂∂x i 2 + y i 2∂∂y + ∑ k 1y i2 ,k 1∂⎞ ⎛⎠∂y k1⎝Y x i 1 + ∑ k 1[δ k 1i 1Y y − X k 1x i 1]y k1 +⎞+ ∑ ] [−δ k 1i 1X k 2y y k1 y k2⎠ − ∑ ][X k 1x + y i 2 i 2X k 1y y i1 ,k 1k 1 ,k 2 k 1(3.10)( ∂=∂x i 2) ⎛ ⎞⎝Y x i 1 + ∑ [δ k 1i 1Y y − X k 1]yx i 1 k1 + ∑ [ ]−δ k 1i 1X k 2y y k1 y k2⎠ +k 1 k 1 ,k 2( ) ⎛ ⎞∂+ y i2⎝Y∂y x i 1 + ∑ [δ k 1i 1Y y − X k 1]yx i 1 k1 + ∑ [ ]−δ k 1i 1X k 2y y k1 y k2⎠+k 1 k 1 ,k 2⎛ ⎞ ⎛+ ⎝ ∑ ∂y i2 ,k 1⎠∂y k1k 1+ ∑ k 1[−X k 1x i 2⎝Y x i 1 + ∑ k 1[δ k 1i 1Y y − X k 1x i 1]y k1 ,i 1+ ∑ k 1[−X k 1y]y i2 y i1 ,k 1⎞]y k1 + ∑ [ ]−δ k 1i 1X k 2y y k1 y k2⎠ +k 1 ,k 2= Y x i 1x i 2 + ∑ [δ k 1i 1Y x i 2y − X k 1]yx i 1x i 2 k1 + ∑ [ ]−δ k 1i 1X k 2yx i 2y k1 y k2 +k 1 k 1 ,k 2+ Y x i 1y y i2 + ∑ [δ k 1i 1Y yy − X k 1]yx i 1y k1 y i2 + ∑ [ ]−δ k 1i 1X k 2yy y k1 y k2 y i2 +k 1 k 1 ,k 2+ ∑ [δ k 1i 1Y y − X k 1]yx i 1 i2 ,k 1+ ∑ [−δ k 1i 1X k 2y]y k2 y i2 ,k 1+ ∑ [ ]−δ k 1i 1X k 2y y k1 y i2 ,k 2+k 1 k 1 ,k 2 k 1 ,k 2+ ∑ [−X k 1]yx i 2 k1 ,i 1+ ∑ [k 1 k 1−X k 1y]y i2 y i1 ,k 1.Some explanations are needed about the computation of the last two termsof line 11, i.e. about the passage from line 7 of (3.10) just above to line 11.

We have to compute:( ) ⎛ ⎞∑ ∂(3.11)y i2,k 1⎝ ∑ [ ]−δ k1i∂y 1Xyk2y k1 y k2⎠ .k1k 1 k 1,k 2This term is of the form( ) ⎛ ⎞∑ ∂(3.12)A k1⎝ ∑[B k1,k∂y2] y k1 y k2⎠ ,k1k 1 k 1,k 2where the terms B k1 ,k 2are independent of the pure first jet variables y x k. Bythe rule of Leibniz for the differentiation of a product, we may write(3.13)( ) ⎛ ⎞∑ ∂A k1⎝ ∑[B k1,k∂y2]y k1 y k2⎠ =k1k 1 k 1,k 2⎛⎞⎛⎞= ∑ [B k1,k 2] y k2⎝ ∑ ∂A k ′1(y k1 ) ⎠ + ∑ [B k1,k∂y2] y k1⎝ ∑ ∂A k ′k 1,k 2 k 1′ k ′ 2(y k2 ) ⎠1∂yk 1,k 2 k 2′ k ′2= ∑ k 1,k 2[B k1,k 2] y k2 A k1 + ∑ k 1,k 2[B k1,k 2] y k1 A k2 .This is how we have written line 11 of (3.10).Next, the first term Y x i 1y y i2 in line 10 of (3.10) is not in a suitable shape.For reasons of harmony and coherence, we must insert it inside a sum of theform ∑ k 1[·] y k1 . Hence, using the Kronecker symbol, we transform:(3.14) Y x i 1y y i2 ≡ ∑ k 1[δk 1i 2Y x i 1y]yk1 .Also, we must “summify” the seven other terms, remaining in lines 10, 11and 12 of (3.10). Sometimes, we use the symmetry y i2 ,k 1≡ y k1 ,i 2withoutmention. Similarly, we get:∑(3.15)[δk 1]i 1Y yy − X k 1x i 1 y yk1 y i2 ≡ ∑ [k 1 k 1 ,k 2∑ ]yk1 y k2 y i2 ≡ ∑[−δk 1i 1X k 2yyk 1 ,k 2∑[ δk 1i 1Y y − X k 1x i 1k 1δ k 1i 1δ k 2i 2Y yy − δ k 2[−δk 1i 1δ k 3i 2X k 2yyk 1 ,k 2 ,k 3i 2X k 1x i 1y]yk1 y k2 y k3 ,]y k1 y k2 ,]yk1 ,i 2≡ ∑ k 1 ,k 2[δk 1i 1δ k 2i 2Y y − δ k 2i 2X k 1x i 1]yk1 ,k 2,∑ [−δk 1]i 1X k 2y yk2 y k1 ,i 2= ∑ [−δk 2]i 1X k 1y yk1 y k2 ,i 2k 1 ,k 2 k 1 ,k 2≡∑ [−δk 2]i 1δ k 3i 2X k 1y yk1 y k2 ,k 3,k 1 ,k 2 ,k 3323

324∑[−δk 1i 1X k 2yk 1 ,k 2∑ [−Xk 1x i 2k 1∑[−Xk 1yk 1]yk1 y k2 ,i 2≡ ∑[−δk 1i 1δ k 3i 2X k 2yk 1 ,k 2 ,k 3]yk1 ,i 1≡ ∑ k 1 ,k 2[−δk 2i 1X k 1x i 2]yk1 ,k 2,]yi2 y k1 ,i 1= ∑ [−Xk 2]y yi2 y k2 ,i 1k 2≡]yk1 y k2 ,k 3,∑ [−δk 1]i 2δ k 3i 1X k 2y yk1 y k2 ,k 3.k 1 ,k 2 ,k 3In the sequel, for products of Kronecker symbols, it will be convenient toadopt the following self-evident contracted notation:(3.16) δ k 1i 1δ k 2i 2≡ δ k 1,k 2i 1 , i 2; generally : δ k 1i 1δ k 2i 2 · · ·δ k λi λ≡ δ k 1,k 2 ,···,k λi 1 , i 2 , ···,i λ.Re-inserting plainly these eight summified terms (3.14), (3.15) in the lastexpression (3.10) of Y i1 ,i 2(lines 10, 11 and 12), we get:(3.17)Y i1 ,i 2= Y x i 1x i + ∑ ]2[δ k 11 i 1Y x i 2y − X k 1yx i 1x i 2 k1k 12+ ∑ ] [−δ k 1i 1X k 2yx i 2y k1 y k2 +k 1 ,k 23+ ∑ k 1[δ k 1i 2Y x i 1y]y k12+ ∑ ][δ k 1,k 2i 1 , i 2Y yy − δ k 2i 2X k 1yx i 1y k1 y k2 +k 1 ,k 23+ ∑ [ ]−δ k 1,k 3i 1 , i 2X k 2yy y k1 y k2 y k3k 1 ,k 2 ,k 34+ ∑ [−δ k 2,k 3i 1 , i 2X k 1yk 1 ,k 2 ,k 3+ ∑ ] [−δ k 2i 1X k 1yx i 2 k1 ,k 2k 1 ,k 25]y k1 y k2 ,k 36+ ∑ ][δ k 1,k 2i 1 , i 2Y y − δ k 2i 2X k 1yx i 1 k1 ,k 2+k 1 ,k 25+ ∑ [ ]−δ k 1,k 3i 1 , i 2X k 2y y k1 y k2 ,k 3+k 1 ,k 2 ,k 36+ ∑ [−δ k 1,k 3i 2 , i 1X k 2yk 1 ,k 2 ,k 3]y k1 y k2 ,k 3.6

Next, we gather the underlined terms, ordering them according to their number.This yields 6 collections of sums of monomials in the pure jet variables:(3.18)Y i1 ,i 2= Y x i 1x i 2 + ∑ ][δ k 1i 1Y x i 2y + δ k 1i 2Y x i 1y − X k 1yx i 1x i 2 k1 +k 1+ ∑ ][δ k 1,k 2i 1 , i 2Y yy − δ k 1i 1X k 2x i 2y − δk 2i 2X k 1yx i 1y k1 y k2 +k 1 ,k 2+ ∑ [ ]−δ k 1,k 3i 1 , i 2X k 2yy y k1 y k2 y k3 +k 1 ,k 2 ,k 3+ ∑ k 1 ,k 2[δ k 1,k 2i 1 , i 2Y y − δ k 2i 2X k 1x i 1 − δk 2i 1X k 1x i 2]y k1 ,k 2++ ∑ []−δ k 2,k 3i 1 , i 2X k 1y − δ k 1,k 3i 1 , i 2X k 2y − δ k 1,k 3i 2 , i 1X k 2y y k1 y k2 ,k 3.k 1 ,k 2 ,k 3To attain the real perfect harmony, this last expression has still to be workedout a little bit.Lemma 3.19. The final expression of Y i1 ,i 2is as follows:(3.20)⎧Y i1 ,i 2= Y x i 1x i 2 + ∑ [δk 1i 1Y x i 2y + δ k 1i 2Y x i 1y − X k 1x i 1x 2] i yk1 +k 1⎪⎨⎪⎩+ ∑ ][δ k 1,k 2i 1 , i 2Y yy − δ k 1i 1X k 2x i 2y − δk 1i 2X k 2yx i 1y k1 y k2 +k 1 ,k 2+ ∑ [ ]−δ k 1,k 2i 1 , i 2X k 3yy y k1 y k2 y k3 +k 1 ,k 2 ,k 3+ ∑ k 1 ,k 2[δ k 1,k 2i 1 , i 2Y y − δ k 1i 1X k 2x i 2 − δk 1i 2X k 2x i 1]y k1 ,k 2++ ∑ []−δ k 1,k 2i 1 , i 2X k 3y − δ k 3,k 1i 1 , i 2X k 2y − δ k 2,k 3i 1 , i 2X k 1y y k1 y k2 ,k 3.k 1 ,k 2 ,k 3Proof. As promised, we explain every tiny detail.The first lines of (3.18) and of (3.20) are exactly the same. For the transformationsof terms in the second, in the third and in the fourth lines, weuse the following device. Let Υ k1 ,k 2be an indexed quantity which is symmetric:Υ k1 ,k 2= Υ k2 ,k 1. Let A k1 ,k 2be an arbitrary indexed quantity. Thenobviously:∑(3.21)A k1 ,k 2Υ k1 ,k 2= ∑ A k2 ,k 1Υ k1 ,k 2.k 1 ,k 2 k 1 ,k 2Similar relations hold with a quantity Υ i1 ,i 2 ,...,i λwhich is symmetric withrespect to its λ indices. Consequently, in the second, in the third and in325

326the fourth lines of (3.18), we may permute freely certain indices in some ofthe terms inside the brackets. This yields the passage from lines 2, 3 and 4of (3.18) to lines 2, 3 and 4 of (3.20).It remains to explain how we pass from the fifth (last) line of (3.18) to thefifth (last) line of (3.20). The bracket in the fifth line of (3.18) contains threeterms: [−T 1 − T 2 − T 3 ]. The term T 3 involves the product δ k 1,k 3i 2 , i 1, which werewrite as δ k 3,k 1i 1 , i 2, in order that i 1 appears before i 2 . Then, we rewrite the threeterms in the new order [−T 2 − T 3 − T 1 ], which yields:(3.22)∑ [−δ k 1,k 3i 1 , i 2X k 2y − δ k 3,k 1i 1 , i 2X k 2y − δ k 2,k 3i 1 , i 2X k 1yk 1 ,k 2 ,k 3]y k1 y k2 ,k 3.It remains to observe that we can permute k 2 and k 3 in the first term −T 2 ,which yields the last line of (3.20). The detailed proof is complete.3.23. Final perfect expression of Y i1 ,i 2 ,i 3. Thanks to similar (longer) computations,we have obtained an expression of Y i1 ,i 2 ,i 3which we consider tobe in final harmonious shape. Without copying the intermediate steps, let uswrite down the result. The comments which are necessary to read it and tointerpret it start just below.Y i1 ,i 2 ,i 3= Y x i 1x i 2x i 3 + ∑ k 1[δ k 1i 1Y x i 2x i 3y + δ k 1i 2Y x i 1x i 3y + δ k 1i 3Y x i 1x i 2y − X k 1x i 1x i 2x i 3]y k1 +(3.24)+ ∑ k 1 ,k 2[δ k 1,k 2i 1 , i 2Y x i 3y 2 + δ k 1,k 2i 3 , i 1Y x i 2y 2 + δ k 1,k 2i 2 , i 3Y x i 1y 2−]−δ k 1i 1X k 2x i 2x i 3y − δk 1i 2X k 2x i 1x i 3y − δk 1i 3X k 2yx i 1x i 2y k1 y k2 ++ ∑ []δ k 1,k 2 ,k 3i 1 , i 2 , i 3Y y 3 − δ k 1,k 2i 1 , i 2X k 3− δ k 1,k 2x i 3y 2 i 1 , i 3X k 3− δ k 1,k 2x i 2y 2 i 2 , i 3X k 3yx i 1y 2 k1 y k2 y k3 +k 1 ,k 2 ,k 3+ ∑ [−δ k 1,k 2 ,k 3i 1 , i 2 , i 3X k 4]yy 3 k1 y k2 y k3 y k4 +k 1 ,k 2 ,k 3 ,k 4+ ∑ k 1 ,k 2[δ k 1,k 2i 1 , i 2Y x i 3y + δ k 1,k 2i 3 , i 1Y x i 2y + δ k 1,k 2i 2 , i 3Y x i 1y −]−δ k 1i 1X k 2x i 2x − i 3 δk 1i 2X k 2x i 1x − i 3 δk 1i 3X k 2yx i 1x i 2 k1 ,k 2++ ∑ [δ k 1,k 2 ,k 3i 1 , i 2 , i 3Y y 2 + δ k 3,k 1 ,k 2i 1 , i 2 , i 3Y y 2 + δ k 2,k 3 ,k 1i 1 , i 2 , i 3Y y 2−k 1 ,k 2 ,k 3−δ k 1,k 2i 1 , i 2X k 3x i 3y − δk 3,k 1i 1 , i 2X k 2x i 3y − δk 2,k 3i 1 , i 2X k 1x i 3y −−δ k 1,k 2i 1 , i 3X k 3x i 2y − δk 3,k 1i 1 , i 3X k 2x i 2y − δk 2,k 3i 1 , i 3X k 1x i 2y −−δ k 1,k 2i 2 , i 3X k 3x i 1y − δk 3,k 1i 2 , i 3X k 2x i 1y − δk 2,k 3i 2 , i 3X k 1x i 1y]y k1 y k2 ,k 3+

+ ∑k 1 ,k 2 ,k 3 ,k 4[−δ k 1,k 2 ,k 3i 1 , i 2 , i 3X k 4y 2 − δ k 2,k 3 ,k 1i 1 , i 2 , i 3X k 4y 2 − δ k 3,k 2 ,k 1i 1 , i 2 , i 3X k 4y 2 −−δ k 3,k 4 ,k 1i 1 , i 2 , i 3X k 2y 2 − δ k 3,k 1 ,k 4i 1 , i 2 , i 3X k 2y 2 − δ k 1,k 3 ,k 4i 1 , i 2 , i 3X k 2+ ∑ [−δ k 1,k 2 ,k 3i 1 , i 2 , i 3X k 4y − δ k 2,k 3 ,k 1i 1 , i 2 , i 3X k 4y − δ k 3,k 1 ,k 2i 1 , i 2 , i 3X k 4yk 1 ,k 2 ,k 3 ,k 4+ ∑k 1 ,k 2 ,k 3[δ k 1,k 2 ,k 3i 1 , i 2 , i 3Y y − δ k 1,k 2i 1 , i 2X k 3x i 3 − δk 1,k 2i 1 , i 3X k 3x i 2 − δk 1,k 2i 2 , i 3X k 3x i 1327]yy 2 k1 y k2 y k3 ,k 4+]y k1 ,k 2y k3 ,k 4+]y k1 ,k 2 ,k 3++ ∑ [−δ k 1,k 2 ,k 3i 1 , i 2 , i 3X k 4y − δ k 4,k 1 ,k 2i 1 , i 2 , i 3X k 3y − δ k 3,k 4 ,k 1i 1 , i 2 , i 3X k 2y − δ k 2,k 3 ,k 4i 1 , i 2 , i 3X k 1yk 1 ,k 2 ,k 3 ,k 43.25. Comments, analysis and induction. First of all, by comparing thisexpression of Y i1 ,i 2 ,i 3with the expression (2.8) of Y 3 , we easily guess a partof the (inductional) dictionary beween the cases n = 1 and the case n 1.For instance, the three monomials [·](y 1 ) 3 , [·] y 1 y 2 and [·] (y 1 ) 2 y 2 in Y 3 arereplaced in Y i1 ,i 2 ,i 3by the following three sums:(3.26) ∑∑∑[·]y k1 y k2 y k3 , [·] y k1 y k2 ,k 3, and [·]y k1 y k2 y k3 ,k 4.k 1 ,k 2 ,k 3 k 1 ,k 2 ,k 3 k 1 ,k 2 ,k 3 ,k 4Similar formal correspondences may be observed for all the monomials ofY 1 , Y i1 , of Y 2 , Y i1 ,i 2and of Y 3 , Y i1 ,i 2 ,i 3. Generally and inductively speaking,the monomial(3.27) [·](y λ1 ) µ1 · · ·(y λd ) µ dappearing in the expression (2.25) of Y κ should be replaced by a certainmultiple sum generalizing (3.26). However, it is necessary to think, to pauseand to search for an appropriate formalism before writing down the desiredmultiple sum.The jet variable y λ1 should be replaced by a jet variable corresponding toa λ 1 -th partial derivative, say y k1 ,...,k λ1, where k 1 , . . ., k λ1 = 1, . . .,n. Forthe moment, to simplify the discussion, we leave out the presence of a sumof the form ∑ k 1 ,...,k λ1. The µ 1 -th power (y λ1 ) µ 1should be replaced not by(y k1 ,...,k λ1) µ1,but by a product of µ1 different jet variables y k1 ,...,k λ1of lengthλ, with all indices k α = 1, . . .,n being distinct. This rule may be confirmedby inspecting the expressions of Y i1 , of Y i1 ,i 2and of Y i1 ,i 2 ,i 3. So y k1 ,...,k λ1should be developed as a product of the form(3.28) y k1 ,...,k λ1y kλ1 +1,...,k 2λ1 · · · y k(µ1 −1)λ 1 +1,...,k µ1 λ 1,where(3.29) k 1 , . . .,k λ1 , . . .,k µ1 λ 1= 1, . . ., n.]y k1 y k2 ,k 3 ,k 4.

328Consider now the product (y λ1 ) µ 1(y λ2 ) µ 2. How should it develope in thecase of several independent variables? For instance, in the expression ofY i1 ,i 2 ,i 3, we have developed the product (y 1 ) 2 y 2 as y k1 y k2 y k3 ,k 4. Thus, areasonable proposal of formalism would be that the product (y λ1 ) µ 1(y λ2 ) µ 2should be developed as a product of the form(3.30)wherey k1 ,...,k λ1y kλ1 +1,...,k 2λ1 · · · y k(µ1 −1)λ 1 +1,...,k µ1 λ 1y kµ1 λ 1 +1,...,k µ1 λ 1 +λ 2· · · y kµ1 λ 1 +(µ 2 −1)λ 2 +1,...,k µ1 λ 1 +µ 2 λ 2,(3.31) k 1 , . . .,k λ1 , . . .,k µ1 λ 1, . . ., k µ1 λ 1 +µ 2 λ 2= 1, . . ., n.However, when trying to write down the development of the general monomial(y λ1 ) µ 1(y λ2 ) µ2 · · ·(y λd ) µ d, we would obtain the complicated product(3.32)y k1 ,...,k λ1y kλ1 +1,...,k 2λ1 · · · y k(µ1 −1)λ 1 +1,...,k µ1 λ 1y kµ1 λ 1 +1,...,k µ1 λ 1 +λ 2...y kµ1 λ 1 +(µ 2 −1)λ 2 +1,...,k µ1 λ 1 +µ 2 λ 2......... ............... ............... ...............y kµ1 λ 1 +···+µ d−1 λ d−1 +1,...,k µ1 λ 1 +···+µ d−1 λ d−1 +λ d· · ·· · · y kµ1 λ 1 +···+µ d−1 λ d−1 +(µ d −1)λ d +1,...,k µ1 λ 1 +···+µ d λ d.Essentially, this product is still readable. However, in it, some of the integersk α have a too long index α, often involving a sum. Such a lengthof α would be very inconvenient in writing down and in reading the generalKronecker symbols δ kα 1 ,...,kα λi 1 ,......,i λwhich should appear in the final expressionof Y i1 ,...,i κ. One should read in advance Theorem 3.73 below to observethe presence of such multiple Kronecker symbols. Consequently, forα = 1, . . ., µ 1 λ 1 , . . .,µ 1 λ 1 + · · · + µ d λ d , we have to denote the indicesk α differently.Notational convention 3.33. We denote d collection of µ d groups of λ d (apriori distinct) integers k α = 1, . . ., n by(3.34)k 1:1:1 , . . .,k} {{ 1:1:λ1 , . . .,k} 1:µ1 :1, . . .,k 1:µ1 :λ} {{ }1λ 1 λ} {{1}µ 1,k 2:1:1 , . . .,k} {{ 2:1:λ2 , . . .,k} 2:µ2 :1, . . .,k 2:µ2 :λ} {{ }2λ 2 λ} {{2}µ 2,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .k d:1:1 , . . .,k} {{ d:1:λd , . . ., k} d:µd :1, . . .,k d:µd :λ} {{ } dλ d.λ dµ d} {{ }

Correspondingly, we identify the set(3.35){1,... ,λ 1 ,... ,µ 1 λ 1 ,...... ,µ 1 λ 1 + µ 2 λ 2 ,... ...,µ 1 λ 1 + µ 2 λ 2 + · · · + µ d λ d }of all integers α from 1 to µ 1 λ 1 + µ 2 λ 2 + · · · + µ d λ d with the followingspecific set(3.36) {1:1:1,... ,1:1:λ} {{ 1 ,...,1:µ} 1 :λ 1 ,... ,2:µ 2 :λ 2 ,... ,d:µ d :λ d },λ}1{{ }µ 1 λ 1} {{ }µ 1 λ 1 +µ 2 λ 2} {{ }µ 1 λ 1 +µ 2 λ 2 +···+µ d λ dwritten in a lexicographic way which emphasizes clearly the subdivision ind collections of µ d groups of λ d integers.With this notation at hand, we see that the development, in several independentvariables, of the general monomial (y λ1 ) µ1 · · ·(y λd ) µ d, may bewritten as follows:(3.37)y k1:1:1 ,...,k 1:1:λ1 · · · y k1:µ1 :1,...,k 1:µ1 :λ 1· · · y kd:1:1 ,...,k d:1:λd · · · · · · y kd:µd :1,...,k d:µd :λ d.Formally speaking, this expression is better than (3.32). Using product symbols,we may even write it under the slightly more compact form∏∏(3.38)y k1:ν1 :1,...,k 1:ν1 :λ 1· · · y kd:νd :1,...,k d:νd :λ d.1ν 1 µ 1 1ν d µ dNow that we have translated the monomial, we may add all the summationsymbols: the general expression of Y κ (which generalizes our three previousexamples (3.26)) will be of the form:(3.39)Y κ = Y x i 1···x iκ +n∑k 1:1:1 ,...,k 1:1:λ1 =1[?]∏∑κ+1d=1· · ·∑1λ 1

3303.40. Combinatorics of the Kronecker symbols. Our next task is to determinewhat appears inside the brackets [?] of the above equation. We willtreat this rather delicate question very progressively. Inductively, we have toguess how we may pass from the bracketed term of (2.25), namely from[ κ · · ·(κ − µ1 λ 1 − · · · − µ d λ d + 1)· Y(λ 1 !) µ 1 µ1 ! · · ·(λ d !) µ d µd ! x κ−µ 1 λ 1 −···−µ d λ d y µ 1 +···+µ d −(3.41) − κ · · ·(κ − µ 1λ 1 − · · · − µ d λ d + 2)(µ 1 λ 1 + · · · + µ d λ d )·(λ 1 !) µ 1 µ1 ! · · ·(λ d !) µ d µd !· X x κ−µ 1 λ 1 −···−µ d λ d +1 y µ 1 +···+µ d −1 ],to the corresponding (still unknown) bracketed term [?].First of all, we examine the following term, extracted from the completeexpression of Y i1 ,i 2 ,i 3(first line of (3.24)):(3.42)n∑k 1 =1[δk 1i 1Y x i 2x i 3y + δ k 1i 2Y x i 1x i 3y + δ k 1i 3Y x i 1x i 2y − X k 1x i 1x i 2x i 3]yk1 .Here, the coefficient [3 Y x 2 y − X x 3] of the monomial y 1 in Y 3 is replacedby the above bracketed terms.Let us precisely analyze the combinatorics. Here, X x 3 is replaced byX k 1x i 1x i 2x , where the lower indices i i 3 1, i 2 , i 3 come from Y i1 ,i 2 ,i 3and wherethe upper index k 1 is the summation index. Also, the integer 3 in 3 Y x 2 y isreplaced by a sum of exactly three terms, each involving a single Kroneckersymbol δi k , in which the lower index is always an index i = i 1 , i 2 , i 3 and inwhich the upper index is always equal to the summation index k 1 . By theway, more generally, we immediately observe that all the successive positiveintegers(3.43) 1, 3, 1, 3, 3, 1, 3, 1, 3, 3, 3, 9, 6, 3, 1, 3, 4appearing in the formula (2.8) for Y 3 are replaced, in the formula (3.24) forY i1 ,i 2 ,i 3, by sums of exactly the same number of terms involving Kroneckersymbols. This observation will be a precious guide. Finally, in the symbolδ k 1i , if i is chosen among the set {i 1 , i 2 , i 3 }, for instance if i = i 1 , it followsthat the development of Y x 2 y necessarily involves the remaining indices, forinstance Y x i 2x i 3y . Since there are three choices for i = i 1 , i 2 , i 3 , we recoverthe number 3.Next, comparing [Y yy − 2 X xy ] (y 1 ) 2 with the termn∑ [](3.44) δ k 1,k 2i 1 , i 2Y yy − δ k 1i 1X k 1x i 2y − δk 1i 2X k 1yx i 1y k1 y k2 ,k 1 ,k 2 =1extracted from the complete expression of Y i1 ,i 2(second line of (3.18)),we learn and we guess that the number of Kronecker symbols before Y x γ y δ

must be equal to the number of indices k α minus γ. This rule is confirmedby examining the term (second and third line of (3.24))∑ [δ k 1,k 2i 1 , i 2Y x i 3y 2 + δ k 1,k 2i 3 , i 1Y x i 2y 2 + δ k 1,k 2i 2 , i 3Y x i 1y 2−(3.45)k 1 ,k 2]−δ k 1i 1X k 2x i 2x i 3y − δk 1i 2X k 2x i 1x i 3y − δk 1i 3X k 2yx i 1x i 2y k1 y k2 ,developing [3 Y xy 2 − 3 X x 2 y] (y 1 ) 2 .Also, we may examine the following term331(3.46)n∑k 1 ,k 2 =1[δ k 1,k 2i 1 , i 2Y x i 3x i 4y 2 + δ k 1,k 2i 1 , i 3Y x i 2x i 4y 2 + δ k 1,k 2i 1 , i 4Y x i 2x i 3y 2++δ k 1,k 2i 2 , i 3Y x i 1x i 4y 2 + δ k 1,k 2i 2 , i 4Y x i 1x i 3y 2 + δ k 1,k 2i 3 , i 4Y x i 1x i 2y 2−−δ k 1i 1X k 1x i 2x i 3x i 4y − δk 1i 2X k 1x i 1x i 2x i 3y − δk 1i 3X k 1x i 1x i 2x i 4y −−δ k 1i 4X k 1x i 1x i 2x i 3y]y k1 y k2 ,extracted from Y i1 ,i 2 ,i 3 ,i 4and developing [6 Y x 2 y 2 − 4 X x 3 y] (y 1 ) 2 . Wewould like to mention that we have not written the complete expression ofY i1 ,i 2 ,i 3 ,i 4, because it would cover two and a half printed pages.By inspecting the way how the indices are permuted in the multiple Kroneckersymbols of the first two lines of this expression (3.46), we observethat the six terms correspond exactly to the six possible choices of two complementaryordered couples of integers in the set {1, 2, 3, 4}, namely(3.47){1, 2} ∪ {3, 4}, {1, 3} ∪ {2, 4}, {1, 4} ∪ {2, 3},{2, 3} ∪ {1, 4}, {2, 4} ∪ {1, 3}, {3, 4} ∪ {1, 2}.At this point, we start to devise the general combinatorics. Before proceedingfurther, we need some notation.3.48. Permutation groups. For every p ∈ N with p 1, we denote by S pthe full permutation group of the set {1, 2, . . ., p − 1, p}. Its cardinal equalsp!. The letters σ and τ will be used to denote an element of S p . If p 2, andif q ∈ N satisfies 1 q p−1, we denote by S q p the subset of permutationsσ ∈ S p satisfying the two collections of inequalities(3.49)σ(1) < σ(2) < · · · < σ(q) and σ(q +1) < σ(q +2) < · · · < σ(p).The cardinal of S q p equals C q p = p!q! (p−q)! .

332Lemma 3.50. For κ 1, the development of (2.20) to several independentvariables (x 1 , . . .,x n ) is:(3.51)⎡⎤n∑Y i1 ,i 2 ,...,i κ= Y x i 1x i 2···x iκ + ⎣ ∑δ k 1i τ(1)Y i x τ(2)···x i τ(κ) y − X k 1 ⎦ yx i 1x i 2 ···x iκ k1 +++n∑k 1 ,k 2 =1n∑⎡k 1 ,k 2 ,k 3 =1+ · · · · · · +++n∑k 1 ,...,k κ=1n∑⎣ ∑τ∈S 2 κ⎡⎣ ∑⎡τ∈S 3 κk 1 ,...,k κ,k κ+1 =1k 1 =1τ∈S 1 κδ k 1, k 2i τ(1) ,i τ(2)Y xi τ(3)···x i τ(κ) y 2 − ∑τ∈S 1 κδ k 1, k 2 , k 3i τ(1) ,i τ(2) ,i τ(3)Y xi τ(4)···x i τ(κ) y 3 −− ∑τ∈S 2 κ⎣δ k 1,...,k κi 1 ,..., i κY y κ −δ k 1i τ(1)X k 2x i τ(2) ···x i τ(κ) yδ k 1, k 2i τ(1),iτ(2)X k 3x i τ(3)···x i τ(κ) y 2 ⎤⎦ y k1 y k2 y k3 +∑τ∈S κ−1κ⎤⎦ y k1 y k2 +δ k 1,......,k κ−1i τ(1) ,...,i τ(κ−1)X kκx i τ(κ) y κ−1 ⎤⎦ y k1 · · · y kκ +[−δ k 1,...,k κi 1 ,..., i κX k κ+1y κ ]y k1 · · · y kκ y kκ+1 + remainder.Here, the term remainder collects all remaining monomials in the pure jetvariables y k1 ,...,k λ.3.52. Continuation. Thus, we have devised how the part of Y i1 ,...,i κwhichinvolves only the jet variables y kα must be written. To proceed further, weshall examine the following term, extracted from Y i1 ,i 2 ,i 3(lines 12 and 13of (3.24))(3.53) ∑ [−δ k 1,k 2 ,k 3i 1 , i 2 , i 3X k 4y− δ k 2,k 3 ,k 12 i 1 , i 2 , i 3X k 4y− δ k 3,k 2 ,k 12 i 1 , i 2 , i 3X k 4y− 2k 1 ,k 2 ,k 3 ,k 4−δ k 3,k 4 ,k 1i 1 , i 2 , i 3X k 2y 2 − δ k 3,k 1 ,k 4i 1 , i 2 , i 3X k 2y 2 − δ k 1,k 3 ,k 4i 1 , i 2 , i 3X k 2y 2 ]y k1 y k2 y k3 ,k 4,which developes the term [−6 X y 2] (y 1 ) 2 y 2 of Y 3 (third line of (2.8)). Duringthe computation which led us to the final expression (3.24), we organizedthe formula in order that, in the six Kronecker symbols, the lower indicesi 1 , i 2 , i 3 are all written in the same order. But then, what is the rule for theappearance of the four upper indices k 1 , k 2 , k 3 , k 4 ?In April 2001, we discovered the rule by inspecting both (3.53) andthe following complicated term, extracted from the complete expression of

Y i1 ,i 2 ,i 3 ,i 4written in one of our manuscripts:∑ [δ k 1,k 2 ,k 3i 1 , i 2 , i 3Y x i 4y 2 + δ k 2,k 1 ,k 3i 1 , i 2 , i 3Y x i 4y 2 + δ k 2,k 3 ,k 1i 1 , i 2 , i 3Y x i 4y 2+k 1 ,k 2 ,k 3(3.54)+ δ k 1,k 2 ,k 3i 1 , i 2 , i 4Y x i 3y 2 + δ k 2,k 1 ,k 3i 1 , i 2 , i 4Y x i 3y 2 + δ k 2,k 3 ,k 1i 1 , i 2 , i 4Y x i 3y 2++ δ k 1,k 2 ,k 3i 1 , i 3 , i 4Y x i 2y 2 + δ k 2,k 1 ,k 3i 1 , i 3 , i 4Y x i 2y 2 + δ k 2,k 3 ,k 1i 1 , i 3 , i 4Y x i 2y 2++ δ k 1,k 2 ,k 3i 2 , i 3 , i 4Y x i 1y 2 + δ k 2,k 1 ,k 3i 2 , i 3 , i 4Y x i 1y 2 + δ k 2,k 3 ,k 1i 2 , i 3 , i 4Y x i 1y 2−− δ k 1,k 2i 1 , i 2X k 3x i 3x i 4y − δk 2,k 1i 1 , i 2X k 3x i 3x i 4y − δk 2,k 3i 1 , i 2X k 1x i 3x i 4y −− δ k 1,k 2i 1 , i 3X k 3x i 2x i 4y − δk 2,k 1i 1 , i 3X k 3x i 2x i 4y − δk 2,k 3i 1 , i 3X k 1x i 2x i 4y −− δ k 1,k 2i 1 , i 4X k 3x i 2x i 3y − δk 2,k 1i 1 , i 4X k 3x i 2x i 3y − δk 2,k 3i 1 , i 4X k 1x i 2x i 3y −− δ k 1,k 2i 2 , i 3X k 3x i 1x i 4y − δk 2,k 1i 2 , i 3X k 3x i 1x i 4y − δk 2,k 3i 2 , i 3X k 1x i 1x i 4y −− δ k 1,k 2i 2 , i 4X k 3x i 1x i 3y − δk 2,k 1i 2 , i 4X k 3x i 1x i 3y − δk 2,k 3i 2 , i 4X k 1x i 1x i 3y −−δ k 1,k 2i 3 , i 4X k 3x i 1x i 2y − δk 2,k 1i 3 , i 4X k 3x i 1x i 2y − δk 2,k 3i 3 , i 4X k 1x i 1x i 2y]y k1 y k2 ,k 3.This sum developes the term [12 Y xy 2 − 18 X x 2 y] y 1 y 2 of Y 3 (third lineof (2.9)). Let us explain what are the formal rules.In the bracketed terms of (3.53), there are no permutation of the indicesi 1 , i 2 , i 3 , but there is a certain unknown subset of all the permutations of thefour indices k 1 , k 2 , k 3 , k 4 . In the bracketed terms of (3.54), two combinatoricsare present:• there are some permutations of the indices i 1 , i 2 , i 3 , i 4 and• there are some permutations of the indices k 1 , k 2 , k 3 .Here, the permutations of the indices i 1 , i 2 , i 3 , i 4 are easily guessed, sincethey are the same as the permutations which were introduced in §3.48 above.Indeed, in the first four lines of (3.54), we see the four decompositions(3.55){i 1 , i 2 , i 3 }∪{i 4 }, {i 1 , i 2 , i 4 }∪{i 3 }, {i 1 , i 3 , i 4 }∪{i 2 }, {i 2 , i 3 , i 4 }∪{i 1 },of the set {i 1 , i 2 , i 3 , i 4 }, and in the last six lines of (3.54), we see the sixdecompositions(3.56){i 1 , i 2 } ∪ {i 3 , i 4 }, {i 1 , i 3 } ∪ {i 2 , i 4 }, {i 1 , i 4 } ∪ {i 2 , i 3 },{i 2 , i 3 } ∪ {i 1 , i 4 }, {i 2 , i 4 } ∪ {i 1 , i 3 }, {i 3 , i 4 } ∪ {i 1 , i 2 },so that (3.54) may be written under the form(3.57) ⎡∑⎣ ∑ ∑δ k τ(1),k τ(2) ,k τ(3)i τ(1) ,i τ(2) ,i τ(3)Y i − ∑x τ(4) y 2k 1 ,k 2 ,k 3 σ∈?τ∈S 3 4τ∈S 2 4∑σ∈?δ k τ(1),k τ(2)i τ(1) ,i τ(2)X k τ(3)333x i τ(3) x i τ(4) y⎤⎦ y k1 y k2 ,k 3,

334where in the two above sums ∑ σ∈?, the letter σ denotes a permutation ofthe set {1, 2, 3} and where the sign ? refers to two (still unknown) subset ofthe full permutation group S 3 . The only remaining question is to determinehow the indices k α are permuted in (3.53) and in (3.54).The answer may be guessed by looking at the permutations of the set{k 1 , k 2 , k 3 , k 4 } which stabilize the monomial y k1 y k2 y k3 ,k 4in (3.53): weclearly have the following four symmetry relations between monomials:(3.58) y k1 y k2 y k3 ,k 4≡ y k2 y k1 y k3 ,k 4≡ y k1 y k2 y k4 ,k 3≡ y k2 y k1 y k4 ,k 3,and nothing more. Then the number 6 of bracketed terms in (3.53) is exactlyequal to the cardinal 24 = 4! of the full permutation group of the set{k 1 , k 2 , k 3 , k 4 } divided by the number 4 of these symmetry relations. Theset of permutations σ of {1, 2, 3, 4} satisfying these symmetry relations(3.59) y kσ(1) y kσ(2) y kσ(3) ,k σ(4)≡ y k1 y k2 y k3 ,k 4consitutes a subgroup of S 4 which we will denote by H (2,1),(1,2)4 . Furthermore,the coset(3.60) F (2,1),(1,2)4 := S 4 /H (2,1),(1,2)4possesses the six representatives( ) ( )1 2 3 4 1 2 3 4,,1 2 3 4 2 3 1 4(3.61) ( ) ( )1 2 3 4 1 2 3 4,,3 4 1 2 3 1 4 2( )1 2 3 4,3 2 1 4( )1 2 3 4,1 3 4 2which exactly appear as the permutations of the upper indices of our example(3.53). Of course, the question arises whether the choice of such sixrepresentatives in the quotient S 4 /H (2,1),(1,2)4 is legitimate.Fortunately, we observe that after conjugation by any permutation σ ∈H (2,1),(1,2)4 , we do not perturb any of the six terms of (3.53), for instance thethird term of (3.53) is not perturbed, as shown by the following computation(3.62)∑ [−δ k σ(3),k σ(2) ,k σ(1)i 1 , i 2 , i 3X k σ(4)]yy 2 k1 y k2 y k3 ,k 4=k 1 ,k 2 ,k 3 ,k 4= ∑ [−δ k 3,k 2 ,k 1i 1 , i 2 , i 3X k σ(4)]yy 2 kσ −1 (1)y kσ −1 (2)y kσ −1 (3) ,k σ −1 (4)k 1 ,k 2 ,k 3 ,k 4= ∑ [−δ k 3,k 2 ,k 1i 1 , i 2 , i 3X k σ(4)]yy 2 k1 y k2 y k3 ,k 4k 1 ,k 2 ,k 3 ,k 4thanks to the symmetry (3.59). Thus, as expected, the choice of 6 arbitraryrepresentatives σ ∈ F (2,1),(1,2)4 in the bracketed terms of (3.53) is free. In

335conclusion, we have shown that (3.53) may be written under the form:⎤∑ ⎢(3.63) ⎣−∑δ k σ(1),k σ(2) ,k σ(3) ⎥⎦y k1 y k2 y k3 ,k 4,k 1 ,k 2 ,k 3 ,k 4⎡σ∈F (2,1),(1,2)4i 1 , i 2 , i 3X k σ(4)y 2This rule is confirmed by inspecting (3.54) (as well as all the otherterms of Y i1 ,i 2 ,i 3and of Y i1 ,i 2 ,i 3 ,i 4). Indeed, the permutations σ of the set{k 1 , k 2 , k 3 } which stabilize the monomial y k1 y k2 ,k 3consist just of the identitypermutation and the transposition of k 2 and k 3 . The coset S 3 /H (1,1),(1,2)3has the three representatives( )1 2 3(3.64),1 2 3(1 2 32 1 3),(1 2 32 3 1which appear in the upper index position of each of the ten lines of (3.54).It follows that (3.54) may be written under the form(3.65)⎡∑⎢∑⎣k 1 ,k 2 ,k 3τ∈S 3 4− ∑ σ∈S 2 4∑σ∈F (1,1),(1,2)3∑τ∈F (1,1),(1,2)3δ k σ(1),k σ(2) ,k σ(3)i τ(1) ,i τ(2) ,i τ(3)δ k σ(1),k σ(2)i τ(1) ,i τ(2)X k σ(3)Y xi τ(4) y 2 −x i τ(3) x i τ(4) y⎤),⎥⎦y k1 y k2 ,k 3.3.66. General complete expression of Y i1 ,...,i κ. As in the incomplete expression(3.39) of Y i1 ,...,i κ, consider integers 1 λ 1 < · · · < λ d κ andµ 1 1, . . ., µ d 1 satisfying µ 1 λ 1 +· · ·+µ d λ d κ+1. By H µ1 λ 1 +···+H µd λ d,we denote the subgroup of permutations τ ∈ S µ1 λ 1 +···+H µd λ dthat leave unchangedthe general monomial (3.38), namely that satisfy∏y kσ(1:ν1 :1),...,k · · ·∏σ(1:ν1 :λ 1y) kσ(d:νd :1),...,k =σ(d:νd :λ d )1ν 1 µ 1 1ν d µ d(3.67)= ∏∏y k1:ν1 :1,...,k 1:ν1 :λ 1· · · y kd:νd :1,...,k d:νd :λ d.1ν 1 µ 1 1ν d µ dThe structure of this group may be described as follows. For every e =1, . . ., d, an arbitrary permutation σ of the set(3.68){e: 1:1, . . ., e: 1:λ } {{ } e , e: 2:1, . . ., e: 2:λ e , · · · , e: µ } {{ } e :1, . . .,e: µ e :λ } {{ } eλ eλ eλ} {{e}µ e}

336which leaves unchanged the monomial(3.69)∏y kσ(e:νe:1) ,...,k σ(e:νe:λe)=1ν eµ e∏1ν eµ ey ke:νe:1 ,...,k e:νe:λe.uniquely decomposes as the composition of• µ e arbitrary permutations of the µ e groups of λ e integers {e : ν e :1, . . ., e:ν e :λ e }, of total cardinal (λ e !) µe ;• an arbitrary permutation between these µ e groups, of total cardinal µ e !.Consequently( )(3.70) Card H (µ 1,λ 1 ),...,(µ d ,λ d )µ 1 λ 1 +···+µ d λ d= µ 1 !(λ 1 !) µ1 · · ·µ d !(λ d !) µ d.Finally, define the coset(3.71) F (µ 1,λ 1 ),...,(µ d ,λ d )µ 1 λ 1 +···+µ d λ d:= S µ1 λ 1 +···+µ d λ d/H (µ 1,λ 1 ),...,(µ d ,λ d )µ 1 λ 1 +···+µ d λ dwith(3.72)( )Card F (µ 1,λ 1 ),...,(µ d ,λ d )µ 1 λ 1 +···+µ d λ d= Card (S µ 1 λ 1 +···+µ d λ d)(Card)H (µ 1,λ 1 ),...,(µ d ,λ d )µ 1 λ 1 +···+µ d λ d= (µ 1λ 1 + · · · + µ d λ d )!.µ 1 !(λ 1 !) µ 1 · · ·µd !(λ d !) µ dIn conclusion, by means of this formalism, we may write down the completegeneralization of Theorem 2.24 to several independent variables.

337Theorem 3.73. For every κ 1 and for every choice of κ indices i 1 , . . .,i κin the set {1, 2, . . ., n}, the general expression of Y i1 ,...,i κis as follows:(3.74)κ+1∑Y i1 ,...,i κ= Y x i 1 ···x iκ +n∑d=1· · ·∑1λ 1

338Appying the chain rule, we may compute:(3.76)h i1 = f 1 [g i1 ] ,h i1 ,i 2= f 2 [g i1 g i2 ] + f 1 [g i1 ,i 2],h i1 ,i 2 ,i 3= f 3 [g i1 g i2 g i3 ] + f 2 [g i1 g i2 ,i 3+ g i2 g i1 ,i 3+ g i3 g i1 ,i 2] + f 1 [g i1 ,i 2 ,i 3],h i1 ,i 2 ,i 3 ,i 4= f 4 [g i1 g i2 g i3 g i4 ] + f 3 [g i2 g i3 g i1 ,i 4+ g i3 g i1 g i2 ,i 4+ g i1 g i2 g i3 ,i 4++g i1 g i4 g i2 ,i 3+ g i2 g i4 g i1 ,i 3+ g i3 g i4 g i1 ,i 2]++ f 2 [g i1 ,i 2g i3 ,i 4+ g i1 ,i 3g i2 ,i 4+ g i1 ,i 4g i2 ,i 3] ++ f 2 [g i1 g i2 ,i 3 ,i 4+ g i2 g i1 ,i 3 ,i 4+ g i3 g i1 ,i 2 ,i 4+ g i4 g i1 ,i 2 ,i 3] ++ f 1 [g i1 ,i 2 ,i 3 ,i 4] .Introducing the derivations(3.77)n∑Fi 2 := g k1 ,iF 3i :=k 1 =1n∑k 1 =1k 1 =1g k1 ,i(∂+ g i f 2∂g k1∂∂g k1+n∑k 1 ,k 2 =1)∂,∂f 1g k1 ,k 2 ,i(∂+ g i f 2∂g k1 ,k 2∂∂f 1+ f 3)∂,∂f 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .n∑Fi λ ∂n∑ ∂:= g k1 ,i + g k1 ,k∂g 2 ,i + · · · +k1 ∂g k1 ,k 2+n∑k 1 ,...,k λ−1 =1+ g i(f 2k 1 ,k 2 =1∂∂f 1+ f 3g k1 ,...,k λ−1 ,i∂∂g k1 ,...,k λ−1+∂∂f 2+ · · · + f λ)∂,∂f λ−1we observe that the following induction relations hold:(3.78)h i1 ,i 2= F 2i 2(h i1 ),h i1 ,i 2 ,i 3= F 3i 3(h i1 ,i 2) ,. . . . . . . . . . . . . . . . . . . . .h i1 ,i 2 ,...,i λ= F λi λ(hi1 ,i 2 ,...,i λ−1).To obtain the explicit version of the Faà di Bruno in the case of severalvariables (x 1 , . . ., x n ) and one variable y, it suffices to extract from the expressionof Y i1 ,...,i κprovided by Theorem 3.73 only the terms correspondingto µ 1 λ 1 + · · ·+µ d λ d = κ, dropping all the X terms. After some simplificationsand after a translation by means of an elementary dictionary, we obtaina statement.

Theorem 3.79. For every integer κ 1 and for every choice of indicesi 1 , . . .,i κ in the set {1, 2, . . ., n}, the κ-th partial derivative of the compositefunction h = h(x 1 , . . .,x n ) = f(g(x 1 , . . .,x n )) with respect to thevariables x i 1, . . .,x iκ may be expressed as an explicit polynomial dependingon the derivatives of f, on the partial derivatives of g and having integercoefficients:(3.80)∂ κ h∂x i 1 · · ·∂xi κ=⎡⎢⎣κ∑d=1∑1λ 1

340The induction formulas are:(4.4)⎧⎪⎨Y j 1 := D 1 ( Y j) − D 1 (X ) y j 1,Y j 2 := D2 ( Y j 1)− D 1 (X ) y j 2 ,· · · · · · · · · · · ·⎪⎩Y j λ := ( Dλ Yλ−1) j − D 1 (X )y j λ ,where the total differentiation operators D λ are denoted by (insteadof (1.22)):(4.5) D λ := ∂∂x + m∑l=1y l 1∂∂y + ∑ m ll=1∂y2l ∂y1l+ · · · +m∑l=1∂yλl ∂yλ−1l.Applying the definitions in the first two lines of (4.4), we compute, we simplifyand we organize the results in a harmonious way, using in an essentialway the Kronecker symbol. Here, the computations are more elementarythan the computations of Y i1 and of Y i1 ,i 2achieved thoroughly in the previousSection 3, so that we do not provide a Latex track of the details. Firstlyand secondly:(4.6) ⎧⎪⎨⎪⎩Y j 1 = Y jx +Y j 2 = Y jx 2 +Thirdly:(4.7)Y j 3 = Y jx 3 +m∑l 1 =1m∑l 1 =1[Y jy l 1 − δj l 1X x]y l 11 +m∑l 1 ,l 2 =1[]2Y jxy − l 1 δj l 1X x 2 y l 11 +[−δ j l 1X y l 2]y l 11 y l 21 ,m∑l 1 ,l 2 =1[Y jy l 1y l 2 − δj l 12X xy l 2]y l 11 y l 21 ++ ∑ [−δ j l 1X y l 2y l 3]y l 11 y l 21 y l 31 + ∑ [ ]Y jy − l 1 δj l 12X x y l 12 +l 1 ,l 2 ,l 3 l 1m∑]+[−δ j l 1X y l 2 − δ j l 22X y l 1 y l 11 y l 22 .l 1 ,l 2 =1m∑l 1 =1[]3Y jx 2 y − l 1 δj l 1X x 3 y l 11 +m∑l 1 ,l 2 =1+ ∑l 1 ,l 2 ,l 3[Y jy l 1y l 2y l 3 − δj l 13X xy l 2y l 3]y l 11 y l 21 y l 31 ++ ∑l 1 ,l 2 ,l 3 ,l 4[−δ j l 1X y l 2y l 3y l 4]y l 11 y l 21 y l 31 y l 41 +[]3Y jxy l 1y − l 2 δj l 13X x 2 y l 2 y l 11 y l 21 +m∑l 1 =1[3Y jxy l 1 − δj l 13X x 2]y l 12 +

+++m∑l 1 ,l 2 =1m∑l 1 ,l 2 ,l 3 =1m∑l 1 =1[3Y jy l 1y l 2 − δj l 13X xy l 2 − δ j l 26X xy l 1]y l 11 y l 22 +[−δ j l 13X y l 2y l 3 − δ j l 33X y l 1y l 2]y l 11 y l 21 y l 32 +[Y jy l 1 − δj l 13X x]y l 13 +m∑l 1 ,l 2 =1m∑l 1 ,l 2 =1[−δ j l 1X y l 2 − δ j l 23X y l 1]y l 11 y l 23 .341[−δ j l 33X y l 2]y l 12 y l 22 +Fourthly:Y j 4 = Y jx 4 ++++m∑l 1 =1m∑l 1 ,l 2 ,l 3 =1m∑l 1 ,l 2 ,l 3 ,l 4 =1m∑l 1 ,l 2 ,l 3 ,l 4 ,l 5 =1[]4Y jx 3 y − l 1 δj l 1X x 4 y l 11 +m∑l 1 ,l 2 =1[]4Y jxy l 1y l 2y − l 3 δj l 16X x 2 y l 2y l 3 y l 11 y l 21 y l 31 +[Y jxy l 1y l 2y l 3y l 4 − δj l 14X xy l 2y l 3y l 4]y l 11 y l 21 y l 31 y l 41 +[−δ j l 1X y l 2y l 3y l 4y l 5]y l 11 y l 21 y l 31 y l 41 y l 51 +[]6Y jx 2 y l 1y − l 2 δj l 14X x 3 y l 2 y l 11 y l 21 +m∑l 1 =1[]6Y jx 2 y − l 1 δj l 14X x 3 y l 12 +(4.8)+++m∑l 1 ,l 2 =1m∑l 1 ,l 2 ,l 3 =1m∑l 1 ,l 2 ,l 3 ,l 4 =1[]12Y jxy l 1y − l 2 δj l 16X x 2 y l 2 − δ j l 212X x 2 y l 1 y l 11 y l 22 +[6Y jy l 1y l 2y l 3 − δj l 112X xy l 2y l 3 − δ j l 312X xy l 1y l 2]y l 11 y l 21 y l 32 +[−δ j l 16X y l 2y l 3y l 4 − δ j l 44X y l 1y l 2y l 3]y l 11 y l 21 y l 31 y l 42 +

342++++++m∑l 1 ,l 2 =1m∑l 1 ,l 2 ,l 3 =1m∑l 1 ,l 2 =1m∑l 1 ,l 2 ,l 3 =1m∑l 1 ,l 2 =1m∑l 1 =1[3Y jy l 1y l 2 − δj l 112X xy l 2]y l 12 y l 22 +[−δ j l 13X y l 2y l 3 − δ j l 212X y l 1y l 3]y l 11 y l 22 y l 32 +[4Y jy l 1y l 2 − δj l 14X xy l 2 − δ j l 212X xy l 1]y l 11 y l 23 +[−δ j l 14X y l 2y l 3 − δ j l 36X y l 1y l 2]y l 11 y l 21 y l 33 +[−δ j l 14X y l 2 − δ j l 26X y l 1]y l 12 y l 23 +[Y jy l 1 − δj l 14X x]y l 14 +m∑l 1 ,l 2 =1m∑l 1 =1[−δ j l 1X y l 2 − δ j l 24X y l 1]y l 11 y l 24 .[]4Y jxy − l 1 δj l 16X x 2 y l 13 +4.9. Inductive elaboration of the general formula. Now we compare theformula (2.9) for Y 4 with the above formula (4.8) for Y j 4 . The goal isto find the rules of transformation and of development by inspecting severalinstances, in order to devise how to transform and to develope the formula(2.25) to several dependent variables.First of all, we have to develope the general monomial (y λ1 ) µ1 · · ·(y λd ) µ d .In every monomial present in the expressions of Y j 1 , of Yj 2 , of Yj 3 and of∑Y j 4 above, we see that the number α of indices l β appearing in all the sumsml 1 ,...,l is exactly equal to µ α=1 1 + · · · + µ d . To denote these µ 1 + · · · + µ dindices l β , we shall use the notation:(4.10) l 1:1 , . . .,l 1:µ1 , . . .,l d:1 , . . .,l d:µd} {{ } } {{ }µ 1 µ} {{d}µ 1 +···+µ d,inspired by Convention 3.33. With such a choice of notation, we may avoidlong subscripts in the indices l β , like l µ1 +···+µ d. It follows that the developmentof the general monomial (y λ1 ) µ1 · · ·(y λd ) µ d to several dependent variablesyields m µ 1+···+µ d possible choices:∏∏(4.11)y l 1:ν 1λ 1· · · · · · y l d:ν dλ d,1ν 1 µ 1 1ν d µ dwhere the indices l 1:1 , . . .,l 1:µ1 , . . .,l d:1 , . . .,l d:µd take their values in the set{1, 2, . . ., m}. Consequently, the general expression of Yκ j must be of the

343form:(4.12)κ+1Yκ j = Y jx κ + ∑d=1m∑l 1:1 =1∑1λ 1

344Then we examine four instances extracted from the complete expression ofY4:j ⎧m∑ []4Y jxy l 1y l 2y − l 3 δj l 16X x 2 y l 2y l 3 y l 11 y l 21 y l 31 ,(4.15)⎪⎨⎪⎩l 1 ,l 2 ,l 3 =1m∑l 1 ,l 2 =1m∑l 1 ,l 2 ,l 3 ,l 4 =1m∑l 1 ,l 2 ,l 3 =1[]12Y jxy l 1y − l 2 δj l 16X x 2 y l 2 − δ j l 212X x 2 y l 1 y l 11 y l 22 ,[−δ j l 16X y l 2y l 3y l 4 − δ j l 44X y l 1y l 2y l 3]y l 11 y l 21 y l 31 y l 42 ,[−δ j l 14X y l 2y l 3 − δ j l 36X y l 1y l 2]y l 11 y l 21 y l 33 ,and we compare them to the corresponding terms of Y 4 :⎧ [ ]4Yxy 3 − 6X x 2 y 2 (y1 ) 3 ,(4.16)⎪⎨⎪⎩[ ]12Yxy 2 − 18X x 2 y y1 y 2 ,[ ]−10Xy 3 (y1 ) 3 y 2 ,[ ]−10Xy 2 (y1 ) 2 y 3 .In the development from (4.16) to (4.15), we see that the four integers justbefore X , namely 6 = 6, 18 = 6 + 12, 10 = 6 + 4 and 10 = 4 + 6, aresplit in a certain manner. Also, a single Kronecker symbol δ j l αis added as afactor. What are the rules?In the second splitting 18 = 6 + 12, we see that the relative weight of6 and of 12 is the same as the relative weight of 1 and 2 in the splitting3 = 1 + 2 issued from the lower indices of the corresponding monomialy l 11 y l 22 . Similarly, in the third splitting 10 = 6 + 4, the relative weight of6 and of 4 is the same as the relative weight of 1 + 1 + 1 and of 2 issuedfrom the lower indices of the corresponding monomial y l 11 y l 21 y l 31 y l 42 . Thisrule may be confirmed by inspecting all the other monomials of Y 2 , Y2,jof Y 3 , Y j 3 and of Y 4 , Y4. j For a general κ 1, the splitting of integersjust amounts to decompose the sum appearing inside the brackets of (4.14)as µ 1 λ 1 , µ 2 λ 2 , . . .,µ d λ d . In fact, when we wrote (4.14), we emphasized inadvance the decomposable factor (µ 1 λ 1 + · · · + µ d λ d ).Next, we have to determine what is the subscript α in the Kronecker symbolδ j l α. We claim that in the four instances (4.15), the subscript α is intrinsicallyrelated to weight splitting. Indeed, recall that in the second lineof (4.15), the number 6 of the splitting 18 = 6 + 12 is related to the number1 in the splitting 3 = 1 + 2 of the lower indices of the monomial y l 11 y l 22 . Itfollows that the index l α must be the index l 1 of the monomial y l 11 . Similarly,also in the second line of (4.15), the number 12 of the splitting 18 = 6 + 12

eing related to the number 2 in the splitting 3 = 1 + 2 of the lower indicesof the monomial y l 11 y l 22 , it follows that the index l α attached to the second Xterm must be the index l 2 of the monomial y l 22 .This rule is still ambiguous. Indeed, let us examine the third line of (4.15).We have the splitting 10 = 6 + 4, homologous to the splitting of relativeweights 5 = (1+1+1)+2 in the monomial y l 11 y l 21 y l 31 y l 42 . Of course, it is clearthat we must choose the index l 4 for the Kronecker symbol associated to thesecond X term −4 X y 3, thus obtaining −δ j l 44 X y l 1y l 2y l 3 . However, sincethe monomial y l 11 y l 21 y l 31 has three indices l 1 , l 2 and l 3 , there arises a question:what index l α must we choose for the Kronecker symbol δ j l αattached to thefirst X term 6 X y 3: the index l 1 , the index l 2 or the index l 3 ?The answer is simple: any of the three indices l 1 , l 2 or l 3 works. Indeed,since the monomial y l 11 y l 21 y l 31 is symmetric with respect to all permutationsof the set of three indices {l 1 , l 2 , l 3 }, we have(4.17)∑ m ]m∑][−δ j l 16X y l 2y l 3y l 4 y l 11 y l 21 y l 31 y l 42 =[−δ j l 26X y l 1y l 3y l 4 y l 11 y l 21 y l 31 y l 42 =l 1 ,l 2 ,l 3 ,l 4 =1=l 1 ,l 2 ,l 3 ,l 4 =1m∑l 1 ,l 2 ,l 3 ,l 4 =1345[−δ j l 36X y l 1y l 2y l 4]y l 11 y l 21 y l 31 y l 42 .In fact, we have systematically used such symmetries during the intermediatecomputations (not exposed here) which we achieved manually to obtainthe final expressions of Y j 1, of Y j 2, of Y j 3 and of Y j 4. To fix ideas, we havealways choosen the first index. Here, the first index is l 1 ; in the first sum ofline 9 of (4.8), the first index l α for the second weight 12 is l 2 .This rule may be confirmed by inspecting all the monomials of Y j 2, ofY j 3, of Y j 4 (and also of Y j 5, which we have computed in a manuscript, butnot copied in this Latex file).From these considerations, we deduce that for the general formula, theweight decomposition is simply µ 1 λ 1 , . . .,µ d λ d and that the Kronecker symbolδ j α is intrinsically associated to the weights. In conclusion, building oninductive reasonings, we have formulated the following statement.Theorem 4.18. For one independent variable x, for several dependent variables(y 1 , . . ., y m ) and for κ 1, the general expression of the coefficient

346Yκ j of the prolongation (4.3) of a vector field is:(4.19)κ+1Yκ j = Y jx κ + ∑ ∑ ∑m∑l 1:1 =1d=1· · ·1λ 1

Appying the chain rule, we may compute:(4.21)m∑h 1 = f l1 g l 11 ,h 2 =h 3 =h 4 =h 5 =l 1 =1m∑l 1 ,l 2 =1m∑l 1 ,l 2 ,l 3 =1m∑l 1 ,l 2 ,l 3 ,l 4 =1+ 3f l1 ,l 2g l 11 g l 21 +m∑l 1 =1f l1 g l 12 ,f l1 ,l 2 ,l 3g l 11 g l 21 g l 31 + 3m∑l 1 ,l 2 =1m∑l 1 ,l 2 ,l 3 ,l 4 ,l 5 =1+ 15+ 10m∑l 1 ,l 2 =1f l1 ,l 2 ,l 3 ,l 4g l 11 g l 21 g l 31 g l 41 + 6m∑l 1 ,l 2 ,l 3 =1m∑l 1 ,l 2 =1f l1 ,l 2g l 12 g l 22 + 4m∑l 1 ,l 2 =1f l1 ,l 2g l 11 g l 22 +m∑l 1 ,l 2 ,l 3 =1f l1 ,l 2g l 11 g l 23 +f l1 ,l 2 ,l 3 ,l 4 ,l 5g l 11 g l 21 g l 31 g l 41 g l 51 + 10f l1 ,l 2 ,l 3g l 11 g l 22 g l 32 + 10f l1 ,l 2g l 12 g l 23 + 5m∑l 1 ,l 2 =1m∑l 1 ,l 2 ,l 3 =1f l1 ,l 2g l 11 g l 24 +m∑l 1 =1f l1 g l 13 ,f l1 ,l 2 ,l 3g l 11 g l 21 g l 32 +m∑l 1 =1m∑l 1 ,l 2 ,l 3 ,l 4 =1f l1 g l 14 ,f l1 ,l 2 ,l 3g l 11 g l 21 g l 33 +m∑l 1 =1347f l1 ,l 2 ,l 3 ,l 4g l 11 g l 21 g l 31 g l 42 +f l1 g l 15 .Introducing the derivations(4.22)⎛m∑F 2 := g l ∂m∑12∂g l g l 11 1⎝F 3 :=l 1 =1m∑l 1 =1l 1 =1g l 12∂∂g l 11+1+l 2 =1l 1 =1m∑l 1 =1m∑l 2 =1g l 13∂∂g l 12+f l1 ,l 2m∑l 1 =1⎞∂⎠ ,∂f l2⎛g l 11⎝m∑l 2 =1f l1 ,l 2∂∂f l2+m∑l 2 ,l 3 =1f l1 ,l 2 ,l 3⎞∂⎠,∂f l2 ,l 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .m∑F λ := g l ∂m∑12l 1∂g l + g l ∂m∑11 3=1 1 l 1∂g l + · · · + g l ∂11λ=1 2 l 1∂g l +1=1 λ−1⎛⎞m∑ m∑+ g l 1 ⎝∂m∑ ∂m∑∂1 f l1 ,l 2+ f l1 ,l∂f 2 ,l 3+ · · · + f l1 ,ll2 ∂f 2⎠,...,l λ,l2 ,l 3∂f l2 ,...,l λl 2 ,l 3 =1l 2 ,...,l λ =1

348we observe that the following induction relations hold:(4.23)h 2 = F 2 (h 1 ) ,h 3 = F 3 (h 2 ) ,. . . . . . . . . . . . . . .h λ = F λ (h λ−1 ) .To obtain the explicit version of the Faà di Bruno in the case of one variable xand several variables (y 1 , . . ., y m ), it suffices to extract from the expressionof Y j κ provided by Theorem 4.18 only the terms corresponding to µ 1λ 1 +· · · + µ d λ d = κ, dropping all the X terms. After some simplifications andafter a translation by means of an elementary dictionary, we may formulatea statement.Theorem 4.24. For every integer κ 1, the κ-th partial derivative of thecomposite function h = h(x) = f (g 1 (x), . . .,g m (x)) with respect to x maybe expressed as an explicit polynomial depending on the partial derivativesof f, on the derivatives of g and having integer coefficients:(4.25)d κ hκ∑dx = κd=1∑1λ 1

Secondly:(5.3)Y j i 1 ,i 2= Y jx i 1x i 2 + m ∑++++m∑l 1 ,l 2 =1m∑l 1 ,l 2 ,l 3 =1m∑l 1 =1m∑l 1 ,l 2 =1l 1 =1n∑k 1 ,k 2 =1n∑k 1 ,k 2 =1n∑k 1 =1[]δ k 1i 1Y jx i 2y + l 1 δk 1i 2Y jx i 1y − l 1 δj l 1X k 1y l x i 1x i 12 k1+349[]δ k 1,k 2i 1 , i 2Y jy l 1y − l 2 δj l 2δ k 1i 1X k 2x i 2y − l 1 δj l 2δ k 1i 2X k 2y l 1x i 1y l 1 k1y l 2k2+n∑k 1 ,k 2 ,k 3 =1[]−δ j l 3δ k 1,k 2i 1 , i 2X k 3y l y l 1y l 12 k1y l 2k2y l 3k3+[]δ k 1,k 2i 1 , i 2Y jy − l 1 δj l 1δ k 1i 1X k 2x − i 2 δj l 1δ k 1i 2X k 2y l x i 11 k1 ,k 2+n∑k 1 ,k 2 ,k 3 =1[]−δ j l 1δ k 2,k 3i 1 , i 2X k 1y − l 2 δj l 2δ k 3,k 1i 1 , i 2X k 2y − l 1 δj l 2δ k 1,k 2i 1 , i 2X k 3y l y l 11 k1y l 2k2y l 3k3.Thirdly:Y j i 1 ,i 2 ,i 3= Y jx i 1x i 2x i 3 + m ∑++m∑l 1 ,l 2 =1m∑l 1 ,l 2 ,l 3 =1l 1 =1n∑k 1 ,k 2 =1n∑k 1 =1[]δ k 1i 1Y jx i 2x i 3y + l 1 δk 1i 2Y jx i 1x i 3y + l 1 δk 1i 3Y jx i 1x i 2y − l 1 δj l 1X k 1y l x i 1x i 2x i 13 k1+[δ k 1,k 2i 1 , i 2Y jx i 3y l 1y l 2 + δk 1,k 2i 3 , i 1Y jx i 2y l 1y l 2 + δk 1,k 2i 2 , i 3Y jx i 1y l 1y l 2 −n∑k 1 ,k 2 ,k 3 =1]−δ j l 2δ k 1i 1X k 2x i 2x i 3y − l 1 δj l 2δ k 1i 2X k 2x i 1x i 3y − l 1 δj l 2δ k 1i 3X k 2y l x i 1x i 2y l 11 k1y l 2[δ k 1,k 2 ,k 3i 1 , i 2 , i 3Y jy l 1y l 2y − l 3 δj l 3δ k 1,k 2i 1 , i 2X k 3x i 3y l 1y − l 2]−δ j l 3δ k 1,k 2i 1 , i 3X k 3x i 2y l 1y − l 2 δj l 3δ k 1,k 2i 2 , i 3X k 3y l x i 1y l 1y l 12 k1y l 2k2y l 3k3+k2+++m∑l 1 ,l 2 ,l 3 ,l 4 =1m∑l 1 =1n∑k 1 ,k 2 =1n∑k 1 ,k 2 ,k 3 ,k 4 =1[]−δ j l 4δ k 1,k 2 ,k 3i 1 , i 2 , i 3X k 4y l y l 1y l 2y l 13 k1y l 2k2y l 3k3y l 4k4+[δ k 1,k 2i 1 , i 2Y jx i 3y l 1 + δk 1,k 2i 3 , i 1Y jx i 2y l 1 + δk 1,k 2i 2 , i 3Y jx i 1y l 1 −]−δ j l 1δ k 1i 1X k 2x i 2x − i 3 δj l 1δ k 1i 2X k 2x i 1x − i 3 δj l 1δ k 1i 3X k 2y l x i 1x i 12 k1 ,k 2+

350(5.4)+++++m∑l 1 ,l 2 =1m∑l 1 ,l 2 ,l 3 =1m∑l 1 ,l 2 =1m∑l 1 =1m∑l 1 ,l 2 =1n∑k 1 ,k 2 ,k 3 =1n∑k 1 ,k 2 ,k 3 ,k 4 =1n∑k 1 ,k 2 ,k 3 ,k 4 =1n∑k 1 ,k 2 ,k 3 =1n∑k 1 ,k 2 ,k 3 ,k 4 =1[δ k 1,k 2 ,k 3i 1 , i 2 , i 3Y jy l 1y l 2 + δk 3,k 1 ,k 2i 1 , i 2 , i 3Y jy l 1y l 2 + δk 2,k 3 ,k 1i 1 , i 2 , i 3Y jy l 1y l 2 −−δ j l 1δ k 2,k 3i 1 , i 2X k 1x i 3y l 2 − δj l 1δ k 2,k 3i 1 , i 3X k 1x i 2y l 2 − δj l 1δ k 2,k 3i 2 , i 3X k 1x i 1y l 2 −−δ j l 2δ k 3,k 1i 1 , i 2X k 2x i 3y l 1 − δj l 2δ k 3,k 1i 1 , i 3X k 2x i 2y l 1 − δj l 2δ k 3,k 1i 2 , i 3X k 2x i 1y l 1 −−δ j l 2δ k 1,k 2i 1 , i 2X k 3x i 3y l 1 − δj l 2δ k 1,k 2i 1 , i 3X k 3x i 2y l 1 − δj l 2δ k 1,k 2i 2 , i 3X k 3x i 1y l 1]y l 1k1y l 2k2 ,k 3+[−δ j l 3δ k 1,k 2 ,k 3i 1 , i 2 , i 3X k 4y l 1y l 2 − δj l 3δ k 2,k 3 ,k 1i 1 , i 2 , i 3X k 4y l 1y l 2 − δj l 3δ k 3,k 2 ,k 1i 1 , i 2 , i 3X k 4y l 1y l 2 −−δ j l 2δ k 3,k 4 ,k 1i 1 , i 2 , i 3X k 2y l 1y l 3 − δj l 2δ k 3,k 1 ,k 4i 1 , i 2 , i 3X k 2y l 1y l 3 −−δ j l 2δ k 1,k 3 ,k 4i 1 , i 2 , i 3X k 2y l 1y l 3[−δ j l 2δ k 1,k 2 ,k 3i 1 , i 2 , i 3X k 3y − l 1 δj l 2δ k 2,k 4 ,k 1i 1 , i 2 , i 3X k 3y − l 1 δj l 2δ k 4,k 1 ,k 2i 1 , i 2 , i 3X k 3y l 1]y l 1k1y l 2k2y l 3k3 ,k 4+]y l 1k1 ,k 2y l 2k3 ,k 4+[]δ k 1,k 2 ,k 3i 1 , i 2 ,ß 3Y jy − l 1 δj l 1δ k 1,k 2i 1 , i 2X k 3x − i 3 δj l 1δ k 1,k 2i 1 , i 3X k 3x − i 2 δj l 1δ k 1,k 2i 2 , i 3X k 3y l x i 11 k1 ,k 2 ,k 3+[−δ j l 2δ k 1,k 2 ,k 3i 1 , i 2 , i 3X k 4y l 1 − δj l 2δ k 4,k 1 ,k 2i 1 , i 2 , i 3X k 3y l 1 − δj l 2δ k 3,k 4 ,k 1i 1 , i 2 , i 3X k 2y l 1 −]−δ j l 1δ k 2,k 3 ,k 4i 1 , i 2 , i 3X k 1y l 1y l 2 k1y l 2k2 ,k 3 ,k 4.5.5. Final synthesis. To obtain the general formula for Y j i 1 ,...,i κ, we have toachieve the synthesis between the two formulas (3.74) and (4.19). We startwith (3.74) and we modify it until we reach the final formula for Y j i 1 ,...,i κ.We have to add the µ 1 +· · ·+µ d sums ∑ ml 1:1 =1 · · ·∑ml 1:µ1 =1 · · · · · ·∑ml d:1 =1 · · ·∑ml d:µd =1 ,together with various indices l α . About these indices, the only point whichis not obvious may be analyzed as follows.A permutation σ ∈ F (µ 1,λ 1 ),...,(µ d ,λ d )µ 1 λ 1 +···+µ d λ dyields the list:(5.6)σ(1:1:1), . . ., σ(1:1:λ 1 ), . . .σ(1:µ 1 :1), . . .,σ(1:µ 1 :λ 1 ), . . .. . .,σ(d:1:1), . . ., σ(1:1:λ d ), . . .σ(d:µ d :1), . . .,σ(d:µ d :λ d ),In the sixth line of (3.74), the last term σ(d : µ d : λ d ) of the above listappears as the subscript of the upper index k σ(d:µd :λ d ) of the term X k σ(d:µ d :λ d) .According to the formal rules of Theorem 4.19, we have to multiply thepartial derivative of X k σ(d:µ d :λ d) by a certain Kronecker symbol δ j l α. Thequestion is: what is the subscript α and how to denote it?

As explained before the statement of Theorem 4.19, the subscript α isobtained as follows. The term σ(d : µ d : λ d ) is of the form (e : ν d : γ e ), forsome e with 1 e d, for some ν e with 1 ν e µ e and for some γ e with1 γ e λ e . The single pure jet variable351(5.7) y le:νek e:νe:1 ,...,k e:νe:γe ,...,k e:νe:λeappears inside the total monomial(5.8)∏y l 1:ν 1k 1:ν1 :1,...,k 1:ν1· · ·:λ 11ν 1 µ 1∏y l d:ν dk d:νd :1,...,k d:νd,:λ d1ν d µ dplaced at the end of the formula for Y j i 1 ,...,i κ(see in advance formula (5.13)below; this total monomial generalizes to several dependent variables thetotal monomial appearing in the last line of (3.74)). According to the ruleexplained before the statement of Theorem 4.18, the index l α must be equalto l e:νe , since l e:νe is attached to the monomial (5.7). Coming back to theterm σ(d:µ d :λ d ), we shall denote this index by(5.9) l e:νe =: l π(e:νe:γ e) =: l πσ(d:µd :λ d ),where the symbol π denotes the projection from the set(5.10) {1:1:1, . . ., 1:µ 1 :λ 1 , . . .. . .,d:1:1, . . .,d:µ d :λ d }to the set(5.11) {1:1, . . ., 1:µ 1 , . . .,d:1, . . ., d:µ d }simply defined by π(e:ν e :γ e ) := (e:ν e ).In conclusion, by means of this formalism, we may write down the completegeneralization of Theorems 2.24, 3.73 and 4.18 to several independentvariables and several dependent variablesTheorem 5.12. For j = 1, . . ., m, for every κ 1 and for every choice of κindices i 1 , . . .,i κ in the set {1, 2, . . ., n}, the general expression of Y j i 1 ,...,i κ

352is as follows:(5.13)κ+1Y j i 1 ,...,i κ= Y jx i 1 ···x + ∑iκn∑d=1· · ·∑1λ 1

g j i 1 ,...,i λand similarly for h i1 ,...,i λ. For arbitrary indices l 1 , . . .,l λ = 1, . . .,m,∂the partial derivativeλ fwill be abbreviated by f ∂y l 1 ···∂y l λ l 1 ,...,l λ.Appying the chain rule, we may compute:(5.17)m∑ ]h i1 = f l1[g l 1i1,h i1 ,i 2=h i1 ,i 2 ,i 3=h i1 ,i 2 ,i 3 ,i 4=l 1 =1m∑l 1 ,l 2 =1m∑l 1 ,l 2 ,l 3 =1+m∑l 1 =1f l1 ,l 2[g l 1i1g l 2i2]+m∑l 1 ,l 2 ,l 3 ,l 4 =1m∑l 1 ,l 2 ,l 3 =1+++m∑l 1 =1f l1 ,l 2 ,l 3[g l 1i1g l 2i2g l 3i3]+f l1[g l 1i1 ,i 2 ,i 3],m∑l 1 ,l 2 =1m∑l 1 ,l 2 =1m∑l 1 =1Introducing the derivationsF 2i :=F 3i :=n∑f l1[g l 1i1 ,i 2],m∑l 1 ,l 2 =1f l1 ,l 2 ,l 3 ,l 4[g l 1i1g l 2i2g l 3i3g l 4i4]+353f l1 ,l 2[g l 1i1g l 2i2 ,i 3+ g l 1i2g l 2i1 ,i 3+ g l 1i3g l 2i1 ,i 2]+f l1 ,l 2 ,l 3[g l 1i2g l 2i3g l 3i1 ,i 4+ g l 1i3g l 2i1g l 3i2 ,i 4+ g l 1i1g l 2i2g l 3i3 ,i 4++g l 1i1g l 2i4g l 3i2 ,i 3+ g l 1i2g l 2i4g l 3i3 ,i 1+ g l 1i3g l 2i4g l 3i1 ,i 2]+f l1 ,l 2[g l 1i1 ,i 2g l 2i3 ,i 4+ g l 1i1 ,i 3g l 2i2 ,i 4+ g l 1i1 ,i 4g l 2i2 ,i 3]+f l1 ,l 2[g l 1i1g l 2i2 ,i 3 ,i 4+ g l 1i2g l 2i1 ,i 3 ,i 4+ g l 1i3g l 2i1 ,i 2 ,i 4+ g l 1i4g l 2i1 ,i 2 ,i 3]+f l1[g l 1i1 ,i 2 ,i 3 ,i 4].m∑k 1 =1 l 1 =1n∑k 1 =1 l 1 =1+m∑l 1 =1m∑g l 1k1 ,ig l 1k1 ,i⎛g l 1 ⎝ i∂∂g l 1∂∂g l 1m∑l 2 =1k1+k1+f l1 ,l 2m∑l 1 =1n∑⎛g l 1 ⎝ im∑k 1 ,k 2 =1 l 1 =1∂∂f l2+m∑l 2 =1m∑l 2 ,l 3 =1f l1 ,l 2g l 1k1 ,k 2 ,if l1 ,l 2 ,l 3⎞∂⎠ ,∂f l2∂∂g l +1k1 ,k 2⎞∂⎠,∂f l2 ,l 3(5.18) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

354F λi :=n∑m∑k 1 =1 l 1 =1++g l 1k1 ,in∑∂∂g l 1k1+k 1 ,k 2 ,...,k λ−1 =1 l 1 =1m∑l 1 =1⎛g l 1 ⎝ im∑l 2 =1m∑f l1 ,l 2n∑m∑k 1 ,k 2 =1 l 1 =1g k1 ,k 2 ,...,k λ−1 ,i∂∂f l2++ · · · + ∑m∑l 2 ,l 3 =1g l 1k1 ,k 2 ,i∂∂g l + · · · +1k1 ,k 2∂∂g l +1k1 ,...,k λ−1f l1 ,l 2 ,l 3l 2 ,l 3 ,...,l λf l1 ,l 2 ,l 3 ,...,l λ∂∂f l2 ,l 3+⎞∂⎠,∂f l2 ,l 3 ,...,l λwe observe that the following induction relations hold:(5.19)h i1 ,i 2= F 2i 2(h i1 ),h i1 ,i 2 ,i 3= F 3i 3(h i1 ,i 2) ,. . . . . . . . . . . . . . . . . . . . .h i1 ,i 2 ,...,i λ= F λi λ(hi1 ,i 2 ,...,i λ−1).To obtain the explicit version of the Faà di Bruno in the case of severalvariables (x 1 , . . .,x n ) and several variables (y 1 , . . .,y m ), it suffices to extractfrom the expression of Y j i 1 ,...,i κprovided by Theorem 5.12 only theterms corresponding to µ 1 λ 1 + · · · + µ d λ d = κ, dropping all the X terms.After some simplifications and after a translation by means of an elementarydictionary, we obtain the fourth and the most general multivariate Faàdi Bruno formula.Theorem 5.20. For every integer κ 1 and for every choice of indicesi 1 , . . .,i κ in the set {1, 2, . . ., n}, the κ-th partial derivative of the compositefunction(5.21) h = h(x 1 , . . ., x n ) = f ( g 1 (x 1 , . . .,x n ), . . .,g m (x 1 , . . .,x n ) )

with respect to the variables x i 1, . . .,x iκ may be expressed as an explicitpolynomial depending on the partial derivatives of f, on the partial derivativesof the g j and having integer coefficients:(5.22)∂ κ h∂x i 1 · · ·∂xi κ=κ∑d=1∑1λ 1

356III: Systems of second orderTable of contents1. Explicit characterizations of flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356.2. Completely integrable systems of second order ordinary differential equations . . . 361.3. First and second auxiliary systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .370.§1. EXPLICIT CHARACTERIZATIONS OF FLATNESSIn 1883, S. Lie obtained the following explicit characterization of thelocal equivalence of a second order ordinary differential equation (E 1 ):y xx = F(x, y, y x ) to the Newtonian free particle equation with one degreeof freedom Y XX = 0. All the functions are assumed to be analytic.Theorem 1.1. ([Lie1883], pp. 362–365) Let K = R of C. Let x ∈ Kand y ∈ K. A local second order ordinary differential equation y xx =F(x, y, y x ) is equivalent under an invertible point transformation (x, y) ↦→(X(x, y), Y (x, y)) to the free particle equation Y XX = 0 if and only if thefollowing two conditions are satisfied:(i) F yxy xy xy x= 0, or equivalently F is a degree three polynomial in y x ,namely there exist four functions G, H, L and M of (x, y) such that Fcan be written as(1.2) F(x, y, y x ) = G(x, y)+y x·H(x, y)+(y x ) 2·L(x, y)+(y x ) 3·M(x, y);(ii) the four functions G, H, L and M satisfy the following system of twosecond order quasi-linear partial differential equations:(1.3) ⎧0 = −2 G yy + 4 3 H xy − 2 3 L xx+⎪⎨+ 2 (G L)y − 2 G x M − 4 G M x + 2 3 H L x − 4 3 H H y,0 = − 2 3 H yy + 4 3 L xy − 2 M xx +⎪⎩+ 2 G M y + 4 G y M − 2 (H M) x − 2 3 H y L + 4 3 L L x.Open question 1.4. Deduce an explicit necessary and sufficient conditionfor the associated submanifold of solutions y = Π(x, a, b) to be locallyequivalent to Y = B + XA.

Assuming F = F(x, y x ) to be independent of y, or equivalently assumingM (E1 ) to be:(1.5) y = b + Π(x, a),the author has checked that equivalence to Y = B + XA holds if and onlyif two differential rational expressions annihilate:0 = Π x 2 a( ) 44− 6 Π x 2 a 3 Π xa( ) 25+ 15 Π ( ) 2x 2 a 2 Πxa 2( ) 6− 4 Π x 2 a 2 Π xa( ) 35Πxa Πxa Πxa Πxa357(1.6)− Π x 2 a Π xa 4( ) 5+ 10 Π xa 3 Π x 2 a Π xa 2( ) 6− 15 Π ( 3x 2 a Πxa 2)( ) 7andΠxa Πxa Πxa0 = Π x 4 a 2(Πxa) 2− 6 Π x 3 a 2 Π x 2 a(Πxa) 3− 4 Π x 3 a Π x 2 a 2(Πxa) 3− Π x 4 a Π xa 2(Πxa) 3++ 15 Π ( ) 2x 2 a 2 Πx 2 a( ) 4+ 10 Π ( ) 3x 3 a Π x 2 a Π xa 2 Πx( ) 4− 152 a Πxa 2( ) 5.Πxa Πxa ΠxaAs an application, this characterizes local sphericity of a rigid hypersurfacew = ¯w + i Θ(z, ¯z) of C 2 . The answer for a general y = Π(x, a, b), togetherwith a proof, will appear elsewhere.A modern restitution of Lie’s original proof of Theorem 1.1 may be foundin [Me2004]. In this reference, we generalize Theorem 1.1 to several dependentvariables y = (y 1 , y 2 , . . .,y m ). In the present Part III, we will insteadpass to several independent variables x = (x 1 , x 2 , . . ., x n ).Theorem 1.7. Let K = R or C, let n ∈ N, suppose n 2 and consider asystem of completely integrable partial differential equations in n independentvariables x = (x 1 , . . .,x n ) ∈ K n and in one dependent variable y ∈ Kof the form:(1.8) y x j 1x j 2 (x) = F j 1,j 2(x, y(x), yx 1(x), . . .,y x n(x) ) , 1 j 1 , j 2 n,where F j 1,j 2= F j 2,j 1. Under a local change of coordinates (x, y) ↦→(X, Y ) = (X(x, y), Y (x, y)), this system (1.8) is equivalent to the simplest“flat” system(1.9) Y X j 1X j 2 = 0, 1 j 1 , j 2 n,if and only if there exist arbitrary functions G j1 ,j 2, H k 1j 1 ,j 2, L k 1j 1and M k 1ofthe variables (x, y), for 1 j 1 , j 2 , k 1 n, satisfying the two symmetryconditions G j1 ,j 2= G j2 ,j 1and H k 1j 1 ,j 2= H k 1j 2 ,j 1, such that the equation (1.8)

358is of the specific cubic polynomial form:(1.10)n∑y x j 1x j 2 = G j1 ,j 2+k 1 =1for j 1 , j 2 = 1, . . .,n.y x k 1(H k 1j 1 ,j 2+ 1 2 y x j 1 L k 1j 2+ 1 2 y x j 2 L k 1j 1+ y x j 1 y x j 2 M k 1It may seem quite paradoxical and counter-intuitive (or even false?) thatevery system (1.10), for arbitrary choices of functions G j1 ,j 2, H k 1j 1 ,j 2, L k 1j 1and M k 1, is automatically equivalent to Y X j 1X j 2 = 0. However, a stronghidden assumption holds: that of complete integrability. Shortly, this crucialcondition amounts to say that(1.11) D j3(Fj 1 ,j 2)= Dj2(Fj 1 ,j 3),for all j 1 , j 2 , j 3 = 1, . . ., n, where, for j = 1, . . ., n, the D j are the totaldifferentiation operators defined by(1.12) D j := ∂∂x j + y x j∂n∑∂y +l=1F j,l∂.∂y x lThese conditions are non-void precisely when n 2. More concretely,developing out (1.11) when the F j 1,j 2are of the specific cubic polynomialform (1.10), after some nontrivial manual computation, we obtain the complicatedcubic differential polynomial in the variables y x k. Equating to zeroall the coefficients of this cubic polynomial, we obtain four familes (I’), (II’),(III’) and (IV’) of first order partial differential equations satisfied by G j1 ,j 2,H k 1j 1 ,j 2, L k 1j 1and M k 1:{(I’)0 = G j1 ,j 2 ,x j 3 − G j1 ,j 3 ,x j 2 +⎧n∑k 1 =1H k 1j 1 ,j 2G k1 ,j 3−n∑k 1 =10 = δ k 1j 3G j1 ,j 2 ,y − δ k 1j 2G j1 ,j 3 ,y + H k 1j 1 ,j 2 ,x j 3 − Hk 1j 1 ,j 3 ,x j 2 +H k 1j 1 ,j 3G k1 ,j 2.),(II’)⎪⎨⎪⎩+ 1 2 G j 1 ,j 3L k 1j 2− 1 2 G j 1 ,j 2L k 1+ 1 2 δk 1j 1+ 1 2 δk 1j 2+n∑k 2 =1n∑k 2 =1n∑k 2 =1j 3+G k2 ,j 3L k 2j 2− 1 2 δk 1j 1G k2 ,j 3L k 2j 1− 1 2 δk 1j 3H k 1k 2 ,j 3H k 2j 1 ,j 2−n∑k 2 =1n∑k 2 =1n∑k 2 =1H k 1k 2 ,j 2H k 2j 1 ,j 3.G k2 ,j 2L k 2j 3+G k2 ,j 2L k 2j 1+

359(III’) ⎧0 = ∑ (δ k 2)j 3H k σ(1)j 1 ,j 2 ,y − δk σ(2)j 2H k σ(1)j 1 ,j 3 ,y +σ∈S 2+ 1 2 δk σ(2)j 2L k σ(1)− 1j 1 ,x j 32 δk σ(2)j 3L k σ(1)+ j 1 ,x j 2⎪⎨⎪⎩+ 1 2 δk σ(2)j 1+δ k σ(2)j 2L k σ(1)− 1j 2 ,x j 32 δk σ(2)j 1L k σ(1)+ j 3 ,x j 2G j1 ,j 3M k σ(1)− δ k σ(2)j 3G j1 ,j 2M k σ(1)++δ k σ(1),k σ(2)j 1 , j 2n∑+ 1 2 δk σ(1)j 1+ 1 2 δk σ(1)j 2+ 1 2 δk σ(1)j 3k 3 =1n∑k 3 =1n∑k 3 =1n∑k 3 =1G k3 ,j 3M k 3− δ k σ(1),k σ(2)j 1 , j 3n∑H k σ(2)k 3 ,j 3L k 3j 2− 1 2 δk σ(1)j 1H k σ(2)k 3 ,j 3L k 3j 1− 1 2 δk σ(1)j 3H k 3j 1 ,j 2L k σ(2)k 3− 1 2 δk σ(1)j 2n∑k 3 =1n∑k 3 =1n∑k 3 =1k 3 =1G k3 ,j 2M k 3+H k σ(2)k 3 ,j 2L k 3j 3+H k σ(2)k 3 ,j 2L k 3j 1+)H k 3j 1 ,j 3L k σ(2)k 3.(IV’) ⎧0 = ∑ ( 12 δk σ(3),k σ(2)j 3 , j 1L k σ(1)j 2 ,y − 1 2 δk σ(3),k σ(2)j 2 , j 1L k σ(1)j 3 ,y +σ∈S 3⎪⎨⎪⎩+δ k σ(3),k σ(2)j 2 , j 1M k σ(1)− δ k σ(3),k σ(2)x j 3 j 3 , j 1M k σ(1)+x j 2+δ k σ(3),k σ(1)j 2 , j 1n∑k 4 =1+ 1 4 δk σ(1),k σ(3)j 1 , j 3n∑k 4 =1H k σ(2)k 4 ,j 3M k 4− δ k σ(3),k σ(1)n∑j 3 , j 1k 4 =1L k σ(2)k 4L k 4j 2− 1 4 δk σ(1),k σ(3)n∑j 1 , j 2k 4 =1H k σ(2)k 4 ,j 2M k 4+)L k σ(2)k 4L k 4j 3.(These systems (I’), (II’), (III’) and (IV’) should be distinguished from thesystems (I), (II), (III) and (IV) of Theorem 1.7 in [Me2004], although theyare quite similar.) Here, the indices j 1 , j 2 , j 3 , k 1 , k 2 , k 3 vary in {1, 2, . . ., n}.By S 2 and by S 3 , we denote the permutation group of {1, 2} and of{1, 2, 3}. To facilitate hand- and Latex-writing, partial derivatives are denotedas indices after a comma; for instance, G j1 ,j 2 ,x j 3 is an abreviation for∂G j1 ,j 2/∂x j 3. To deduce (I’), (II’), (III’) and (IV’) from equation (1.11), we

360use the fact that every cubic polynomial equation of the form(1.13)0 ≡ A ++n∑k 1 =1n∑B k1 · y x k 1 +n∑n∑k 1 =1 k 2 =1 k 2 =1n∑n∑k 1 =1 k 2 =1C k1 ,k 2· y x k 1 y x k 2 +D k1 ,k 2 ,k 3· y x k 1 y x k 2 y x k 3is equivalent to the annihilation of the following symmetric sums of its coefficients:(1.14) ⎧0 = A,⎪⎨ 0 = B k1 ,0 = C k1 ,k 2+ C k2 ,k 1,⎪⎩0 = D k1 ,k 2 ,k 3+ D k3 ,k 1 ,k 2+ D k2 ,k 3 ,k 1+ D k2 ,k 1 ,k 3+ D k3 ,k 2 ,k 1+ D k1 ,k 3 ,k 2.for all k 1 , k 2 , k 3 = 1, . . .,n.In conclusion, the functions G j1 ,j 2, H k 1j 1 ,j 2, L k 1j 1and M k 1in the statementof Theorem 1.7 are far from being arbitrary: they satisfy the complicatedsystem of first order partial differential equations (I’), (II’), (III’) and (IV’)above.Our proof of Theorem 1.7 is similar to the one provided in [Me2004], inthe case of systems of second order ordinary differential equations, so thatmost steps of the proof will be summarized.In the end of this paper, we will delineate a complicated system of secondorder partial differential equations satisfied by G j1 ,j 2, H k 1j 1 ,j 2, L k 1j 1andM k 1which is the exact analog of the system described in the abstract. Themain technical part of the proof of Theorem 1.7 will be to establish thatthis second order system is a consequence, by linear combinations and bydifferentiations, of the first order system (I’), (II’), (III’) and (IV’).Open question 1.15. Are Theorems 1.1 and 1.7 true under weaker smoothnessassumptions, namely with a C 2 or a W 1,∞locright-hand side ?We refer to [Ma2003] for inspiration and appropriate tools.Open question 1.16. Deduce from Theorem 1.7 an explicit necessary andsufficient condition for the associated submanifold of solutions y = b +Π(x i , a k , b) to be locally equivalent to Y = B + X 1 A 1 + · · · + X n A n .As an application, this would characterize local sphericity of a Levi nondegeneratehypersurface M ⊂ C n+1 with n 2.Generalizing the Lie-Tresse classification would be a great achievement.

Open problem 1.17. For n = 2 establish a complete list of normal forms ofall possible systems (1.?) according to their Lie symmetry group. In case ofsuccess, classify Levi nondegenerate real analytic hypersurfaces of C 3 up tobiholomorphisms.361§2. COMPLETELY INTEGRABLE SYSTEMS OFSECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS2.1. Prolongation of a point transformation to the second order jetspace. Let K = R or C, let n ∈ N, suppose n 2, let x = (x 1 , . . .,x n ) ∈K n and let y ∈ K. According to the main assumption of Theorem 1.7, wehave to consider a local K-analytic diffeomorphism of the form((2.2) xj 1, y ) ↦−→ ( X j (x j 1, y), Y (x j 1, y) ) ,which transforms the system (1.8) to the system Y X i 1X i 2 = 0, 1 j 1 , j 2 n. Without loss of generality, we shall assume that this transformation isclose to the identity. To obtain the precise expression (2.35) of the transformedsystem (1.8), we have to prolong the above diffeomorphism to thesecond order jet space. We introduce the coordinates (x j , y, y x j 1 , y x j 1x j 2 ) onthe second order jet space. Let(2.3) D k := ∂∂x k + y x k∂n∑∂y +l=1y x k x l∂,∂y x lbe the k-th total differentiation operator. According to [Ol1986, BK1989,Ol1995], for the first order partial derivatives, one has the (implicit, compact)expression:⎛ ⎞ ⎛D 1 X 1 · · · D 1 X n −1 ⎛ ⎞D 1 Y(2.4)⎝Y X 1.Y X n⎠ = ⎝⎞. · · · .⎠D n X 1 · · · D n X n⎝.D n Ywhere (·) −1 denotes the inverse matrix, which exists, since the transformation(2.2) is close to the identity. For the second order partial derivatives,again according to [Ol1986, BK1989, Ol1995], one has the (implicit, compact)expressions:⎛ ⎞ ⎛Y X j X 1 D 1 X 1 · · · D 1 X n ⎞−1 ⎛ ⎞D 1 Y X j(2.5) ⎝.⎠ = ⎝. · · · .⎠ ⎝.⎠ ,Y X j X n D n X 1 · · · D n X n D n Y X jfor j = 1, . . ., n. Let DX denote the matrix (D i X j ) 1jn1in, where i is theindex of lines and j the index of columns, let Y X denote the column matrix(Y X i) 1inand let DY be the column matrix (D i Y ) 1in.⎠ ,

362By inspecting (2.5) above, we see that the equivalence between (i), (ii)and (iii) just below is obvious:Lemma 2.6. The following conditions are equivalent:(i) the differential equations Y X j Xk = 0 hold for 1 j, k n;(ii) the matrix equations D k (Y X ) = 0 hold for 1 k n;(iii) the matrix equations DX · D k (Y X ) = 0 hold for 1 k n;(iv) the matrix equations 0 = D k (DX)·Y X −D k (DY ) hold for 1 k n.Formally, in the sequel, it will be more convenient to achieve the explicitcomputations starting from condition (iv), since no matrix inversion at all isinvolved in it.Proof. Indeed, applying the total differentiation operator D k to the matrixequation (2.4) written under the equivalent form 0 = DX · Y X − DY , weget:(2.7) 0 = D k (DX) · Y X + DX · D k (Y X ) − D k (DY ),so that the equivalence between (iii) and (iv) is now clear.2.8. An explicit formula in the case n = 2. Thus, we can start to developeexplicitely the matrix equations(2.9) 0 = D k (DX) · Y X − D k (DY ).In it, some huge formal expressions are hidden behind the symbol D k . Proceedinginductively, we start by examinating the case n = 2 thoroughly. Bydirect computations which require to be clever, we reconstitute some 3 × 3determinants in the four (in fact three) developed equations (2.9). Aftersome work, the first equation is:(2.10)∣∣ ∣ ∣∣∣∣∣ X 10 = y x 1 x 1 · xX 1 1 xX 1 2 y ∣∣∣∣∣ X 2 xX 2 1 xX 2 2 y +Y x 1 Y x 2 Y y∣⎧ ∣⎨ ∣∣∣∣∣ X 1+ y x 1 ·⎩ 2 xX 1 1 xX 1 2 x 1 yX 2 xX 2 1 xX 2 2 x 1 yY x 1 Y x 2 Y x 1 y∣X 1 xX 1 1 xX 1 ∣∣∣∣∣ 2 x 1 x 1X 2 xX 2 1 xX 2 2 x 1 x+ 1Y x 1 Y x 2 Y x 1 x 1∣ ⎫∣∣∣∣∣ X 1 ∣ − x 1 xX 1 1 xX 1 2 y⎬X 2 x 1 xX 2 1 xX 2 2 yY x 1 x 1 Y x 2 Y ∣⎭ +y⎧ ∣ ⎫⎨ ∣∣∣∣∣ X 1+ y x 2 ·⎩ − xX 1 1 x 1 xX 1 1 y⎬X 2 xX 2 1 x 1 xX 2 1 yY x 1 Y x 1 x 1 Y ∣⎭ +y

⎧∣ ⎨X 1 xX 1+ y x 1 y x 1 ·1 xX 2 yy1 ∣∣∣∣∣ ⎩X 2 xX 2∣1 xX 2 2 yy − 2Y x 1 Y x 2 Y yy∣⎧ ∣ ∣⎫ ⎨ ∣∣∣∣∣ X 1+ y x 1 y x 2 ·⎩ −2 xX 1 1 x 1 yXy1 ∣∣∣∣∣ ⎬X 2 xX 2 1 x 1 yXy2 ⎭ +Y x 1 Y x 1 y Y y⎧ ∣ ∣⎫ ⎨ ∣∣∣∣∣ X 1+ y x 1 y x 1 y x 1 ·⎩ − yy X 1 xX 1 2 y ∣∣∣∣∣ ⎬Xyy 2 X 2 xX 2 2 y⎭ +Y yy Y x 2 Y y⎧ ∣⎨ ∣∣∣∣∣ X 1+ y x 1 y x 1 y x 2 ·⎩ − xX 1 1 yy Xy1X 2 xX 2 1 yy Xy2X 1 x 1 yX 1 x 2 X 1 yX 2 x 1 yX 2 x 2 X 2 yY x 1 y Y x 2 Y y∣ ∣∣∣∣∣⎫⎬⎭ +Y x 1 Y yy Y y∣ ∣∣∣∣∣⎫⎬⎭ .This formula and the two next (2.22), (2.23) have been checked by SylvainNeut and Michel Petitot with the help of Maple.2.11. Comparison with the coefficients of the second prolongation ofa vector field. At present, it is useful to make an illuminating digressionwhich will help us to devise what is the general form of the development ofthe equations (2.9). Consider an arbitrary vector field of the form(2.12) L :=j 1 =1n∑k=1X k∂∂x k + Y ∂∂y ,where the coefficients X k and Y are functions of (x i , y). Accordingto [Ol1986, BK1989, Ol1995], there exists a unique prolongation L (2) ofthis vector field to the second order jet space, of the formn∑(2.13) L (2) ∂n∑ n∑ ∂:= L + Y j1 + Y j1 ,j∂y2,x j 1 ∂y x j 1x j 2j 1 =1 j 2 =1where the coefficients Y j1 , Y j1 ,j 2may be computed by means of formulas(3.4) of Section 3(II). In Part II, we obtained the following perfect formulas:(2.14) ⎧⎪⎨⎪⎩Y j1 ,j 2= Y x j 1x j 2 +++n∑n∑k 1 =1n∑k 1 =1 k 2 =1n∑n∑k 1 =1 k 2 =1 k 3 =1y x k 1 · {δ k 1j 1Y x j 2y + δ k 1j 2Y x j 1y − X k 1x j 1x j 2}+y x k 1 y x k 2 ·n∑363{}δ k 1,k 2j 1 , j 2Y yy − δ k 1j 1X k 2x j 2y − δk 1j 2X k 2+x j 1yy x k 1 y x k 2 y x k 3 ·{ }−δ k 1,k 2j 1 , j 2X k 3yy ,

364for j 1 , j 2 = 1, . . ., n. The expression of Y j1 does not matter for us here.Specifying this formula to the the case n = 2 and taking account of thesymmetry Y 1,2 = Y 2,1 we get the following three second order coefficients:(2.15) ⎧Y 1,1 = Y x 1 x 1 + y x 1 · {2 }Y x 1 y − X 1x 1 x + 1 yx 2 · {−X } 2x 1 x + 1+ y x 1 y x 1 · {Y }yy − 2 X 1x 1 y + yx 1 y x 2 · {−2X 2x y}+ 1+ y x 1 y x 1 y x 1 · {−X }yy1 + yx 1 y x 1 y x 2 · {−Xyy} 2 ,Y 1,2 = Y x 1 x 2 + y x 1 · {Y }x 2 y − X 1x⎪⎨1 x + 2 yx 2 · {Y }x 1 y − X 2x 1 x + 2+ y x 1 y x 1 · {−X } 1x 2 y + yx 1 y x 2 · {Yyy − X 1x 1 y − X 2x y}+ 2+ y x 2 y x 2 · {−X 2x y}+ 1+ y x 1 y x 1 y x 2 · {−X 1yy}+ yx 1 y x 2 y x 2 · {−X 2yy},Y 2,2 = Y x 2 x 2 + y x 1 · {−X }x 1 2 x + 2 yx 2 · {2 }Y x 2 y − Xx 2 2 x + 2+ y x 1 y x 2 · {−2 }X 1x⎪⎩2 y + yx 2 y x 2 · {Yyy − 2 X 2x y}+ 2+ y x 1 y x 2 y x 2 · {−X }yy1 + yx 2 y x 2 y x 2 · {−Xyy} 2 .We would like to mention that the computation of Y j1 ,j 2, 1 j 1 , j 2 2,above is easier than the verification of (2.10). Based on the three formulas(2.15), we claim that we can guess the second and the third equations,which would be obtained by developing and by simplifying (2.9), namelywith y x 1 x 2 and with y x 2 x 2 instead of y x 1 x2 in (2.10). Our dictionary to translatefrom the first formula (2.15) to (2.10) may be described as follows.Begin with the Jacobian determinant(2.16)∣∣X 1 xX 1 1 xX 1 2 y ∣∣∣∣∣X 2 xX 2 1 xX 2 2 yY x 1 Y x 2 Y yof the change of coordinates (2.2). Since this change of coordinates is closeto the identity, we may consider that the following Jacobian matrix approximationholds:⎛⎞ ⎛(2.17) ⎝ X1 xX 1 1 xX 1 2 yX 2 xX 2 1 xX 2 ⎠ ∼ 2 y = ⎝ 1 0 0⎞0 1 0 ⎠ .Y x 1 Y x 2 Y y 0 0 1The jacobian matrix has three columns. There are six possible second orderderivatives with respect to the variables (x 1 , x 2 , y), namely(2.18) (·) x 1 x 1, (·) x 1 x 2, (·) x 2 x 2, (·) x 1 y, (·) x 2 y, (·) yy .In the Jacobian determinant (2.16), by replacing any one of the threecolumns of first order derivatives with a column of second order derivatives,we obtain exactly 3 × 6 = 18 possible determinants. For instance, by

eplacing the third column by the second order derivative (·) x 1 y or the firstcolumn by the second order derivative (·) x 1 x1, we get:365(2.19)∣X 1 x 1 X 1 x 2 X 1 x 1 yX 2 x 1 X 2 x 2 X 2 x 1 yY x 1 Y x 2 Y x 1 y∣or∣X 1 x 1 x 1 X 1 x 2 Xy1X 2 x 1 x 1 X 2 x 2 Xy2Y x 1 x 1 Y x 2 Y y∣ .We recover the two determinants appearing in the second line of (2.10). Onthe other hand, according to the approximation (2.17), these two determinantsare essentially equal to(2.20)∣1 0 X 1 x 1 y0 1 X 2 x 1 y0 0 Y x 1 y∣ = Y x 1 yor to∣X 1 x 1 x 1 0 0X 2 x 1 x 1 1 0Y x 1 x 1 0 1 ∣ ∣∣∣∣∣= X 1 x 1 x 1.Consequently, in the second line of (2.10), up to a change to calligraphicletters, we recover the coefficient(2.21) 2 Y x 1 y − X 1x 1 x 1of y x1 in the expression of Y 1,1 in (2.15). In conclusion, we have discoveredhow to pass symbolically from the first equation (2.15) to the equation (2.10)and conversely.Translating the second equation (2.15), we deduce, without any furthercomputation, that the second equation which would be obtained by developing(2.9) in length, is:(2.22)∣ ∣ ∣∣∣∣∣ X 10 = y x 1 x 2 · xX 1 1 xX 1 2 y ∣∣∣∣∣X 2 xX 2 1 xX 2 2 y +Y x 1 Y x 2 Y y⎧⎨+ y x 1 ·⎩∣⎧⎨+ y x 2 ·⎩∣X 1 x 1 X 1 x 2 X 1 x 2 yX 2 x 1 X 2 x 2 X 2 x 2 yY x 1 Y x 2 Y x 2 yX 1 x 1 X 1 x 2 X 1 x 1 yX 2 x 1 X 2 x 2 X 2 x 1 yY x 1 Y x 2 Y x 1 y∣ Xx 1 X 1 x 1 X 1 ∣∣∣∣∣ 2 x 1 x 2X 2 xX 2∣1 xX 2 2 x 1 x+ 2Y x 1 Y x 2 Y x 1 x ∣ 2 ∣∣∣∣∣ X 1 ∣ − x 1 xX 1 2 xX 1 2 yX 2 x 1 xX 2 2 xX 2 2 yY x 1 x 2 Y x 2 Y y∣ ∣∣∣∣∣ X 1 ∣ − xX 1 1 x 1 xX 1 2 yX 2 xX 2 1 x 1 xX 2 2 yY x 1 Y x 1 x 2 Y y⎫⎬∣⎭ +⎫⎬∣⎭ +

366⎧ ∣ ∣⎫ ⎨ ∣∣∣∣∣ X 1+ y x 1 y x 1 ·⎩ − x 2 yX 1 xX 1 2 y ∣∣∣∣∣ ⎬X 2 x 2 yX 2 xX 2 2 y⎭ +Y x 2 y Y x 2 Y y⎧∣ ∣⎨X 1 xX 1+ y x 1 y x 2 ·1 xX 1 2 yy ∣∣∣∣∣ X 1⎩X 2 xX 2∣1 xX 2 x 2 yy −1 yX 1 xX 2 y1 ∣∣∣∣∣ X 2 xY x 1 Y x 2 Y yy∣1 yX 2 xX 2 2 y −Y x 1 y Y x 2 Y y ∣⎫ ⎧ ∣ ∣⎫ X 1 xX 1−1 x 2 yXy1 ∣∣∣∣∣ ⎬ ⎨ ∣∣∣∣∣ X 1 X 2 xX 2∣1 x 2 yXy2 ⎭ + y x 2 y x 2 ⎩ − xX 1 1 x 1 yXy1 ∣∣∣∣∣ ⎬X 2 xX 2 1 x 1 yXy2 ⎭ +Y x 1 Y x 2 y Y y Y x 1 Y x 1 y Y y⎧ ∣ ∣⎫ ⎨ ∣∣∣∣∣ X 1+ y x 1 y x 1 y x 2 ·⎩ − yy X 1 xX 1 2 y ∣∣∣∣∣ ⎬Xyy 2 X 2 xX 2 2 y⎭ +Y yy Y x 2 Y y⎧ ∣ ∣⎫ ⎨ ∣∣∣∣∣ X 1+ y x 1 y x 2 y x 2 ·⎩ − xX 1 1 yy Xy1 ∣∣∣∣∣ ⎬X 2 xX 2 1 yy Xy2 ⎭ .Y x 1 Y yy Y yUsing the third equation (2.15), we also deduce, without any further computation,that the third equation which would be obtained by developing (2.9)in length, is:(2.23) ∣ ∣ ∣ ∣∣∣∣∣ X 10 = y x 2 x 2 · xX 1 1 xX 1 2 y ∣∣∣∣∣ X 1X 2 xX 2 1 xX 2 xX 12 y +1 xX 1 ∣∣∣∣∣ 2 x 2 x 2X 2 xX 2Y x 1 Y x 2 Y y∣1 xX 2 2 x 2 x+ 2Y x 1 Y x 2 Y x 2 x⎧ ∣ ⎫2⎨ ∣∣∣∣∣ X 1+ y x 1 ·⎩ − x 2 xX 1 2 xX 1 2 y⎬X 2 x 1 xX 2 2 xX 2 2 yY x 2 x 2 Y x 2 Y ∣⎭ +y⎧ ∣ ∣ ⎫⎨ ∣∣∣∣∣ X 1+ y x 2 ·⎩ 2 xX 1 1 xX 1 2 x 2 y ∣∣∣∣∣ X 1X 2 xX 2 1 xX 2 2 x 2 yY x 1 Y x 2 Y x∣ − xX 1 1 x 2 xX 1 2 y⎬X 2 xX 2 1 x 2 xX 2 2 y2 y Y x 1 Y x 2 x 2 Y ∣⎭ +y

⎧ ∣ ∣⎫ ⎨ ∣∣∣∣∣ X 1+ y x 1 y x 2 ·⎩ −2 x 2 yX 1 xX 1 2 y ∣∣∣∣∣ ⎬X 2 x 2 yX 2 xX 2 2 y⎭ +Y x 2 y Y x 2 Y y⎧∣ ⎨X 1 xX 1+ y x 2 y x 2 ·1 xX 2 yy1 ∣∣∣∣∣ ⎩X 2 xX 2∣1 xX 2 2 yy − 2Y x 1 Y x 2 Y yy∣⎧ ∣ ∣⎫ ⎨ ∣∣∣∣∣ X 1+ y x 1 y x 2 y x 2 ·⎩ − yy X 1 xX 1 2 y ∣∣∣∣∣ ⎬Xyy 2 X 2 xX 2 2 y⎭ +Y yy Y x 2 Y y⎧ ∣ ∣⎫ ⎨ ∣∣∣∣∣ X 1+ y x 2 y x 2 y x 2 ·⎩ − xX 1 1 yy Xy1 ∣∣∣∣∣ ⎬X 2 xX 2 1 yy Xy2 ⎭ .Y x 1 Y yy Y yX 1 x 1 X 1 x 2 yX 1 yX 2 x 1 X 2 x 2 yX 2 yY x 1 Y x 2 y Y y∣ ∣∣∣∣∣⎫⎬⎭ +3672.24. Appropriate formalism. To describe the combinatorics underlyingformulas (2.10), (2.22) and (2.23), as in [Me2004], let us introduce the followingnotation for the Jacobian determinant:(2.25) ∆(x 1 |x 2 |y) :=∣∣X 1 xX 1 1 xX 1 2 y ∣∣∣∣∣X 2 xX 2 1 xX 2 2 y .Y x 1 Y x 2 Y yHere, in the notation ∆(x 1 |x 2 |y), the three spaces between the two verticallines | refer to the three columns of the Jacobian determinant, and the termsx 1 , x 2 , y in (x 1 |x 2 |y) designate the partial derivatives appearing in eachcolumn. Accordingly, in the following two examples of modified Jacobiandeterminants:(2.26)⎧∆(x 1 x 2 |x 2 |y) :=⎪⎨∣∆(x 1 |x 2 |x 1 y) :=⎪⎩∣X 1 x 1 x 2 X 1 x 2 Xy1X 2 x 1 x 2 X 2 x 2 Xy2Y x 1 x 2 Y x 2 Y yX 1 x 1 X 1 x 2 X 1 x 1 yX 2 x 1 X 2 x 2 X 2 x 1 yY x 1 Y x 2 Y x 1 y∣,∣andwe simply mean which column of first order derivatives is replaced by acolumn of second order derivatives in the original Jacobian determinant.As there are 6 possible second order derivatives (·) x 1 x 1, (·) x 1 x 2, (·) x 1 x y,(·) x 2 x 2, (·) x 2 y and (·) yy together with 3 columns, we obtain 3 × 6 = 18

368possible modified Jacobian determinants:(2.27)⎧⎪⎨⎪⎩∆(x 1 x 1 |x 2 |y) ∆(x 1 |x 1 x 1 |y) ∆(x 1 |x 2 |x 1 x 1 )∆(x 1 x 2 |x 2 |y) ∆(x 1 |x 1 x 2 |y) ∆(x 1 |x 2 |x 1 x 2 )∆(x 1 y|x 2 |y) ∆(x 1 |x 1 y|y) ∆(x 1 |x 2 |x 1 y)∆(x 2 x 2 |x 2 |y) ∆(x 1 |x 2 x 2 |y) ∆(x 1 |x 2 |x 2 x 2 )∆(x 2 y|x 2 |y) ∆(x 1 |x 2 y|y) ∆(x 1 |x 2 |x 2 y)∆(yy|x 2 |y) ∆(x 1 |yy|y) ∆(x 1 |x 2 |yy).Next, we observe that if we want to solve with respect to y x 1 x1 in (2.10),with respect to y x 1 x 2 in (2.22) and with respect to y x 2 x2 in (2.23), we have todivide by the Jacobian determinant ∆(x 1 |x 2 |y). Consequently, we introduce18 new square functions as follows:(2.28)⎧⎪⎨⎪⎩□ 1 x 1 x 1 := ∆(x1 x 1 |x 2 |y)∆(x 1 |x 2 |y)□ 1 x 2 x 2 := ∆(x2 x 2 |x 2 |y)∆(x 1 |x 2 |y)□ 2 x 1 x 1 := ∆(x1 |x 1 x 1 |y)∆(x 1 |x 2 |y)□ 2 x 2 x 2 := ∆(x1 |x 2 x 2 |y)∆(x 1 |x 2 |y)□ 3 x 1 x 1 := ∆(x1 |x 2 |x 1 x 1 )∆(x 1 |x 2 |y)□ 3 x 2 x 2 := ∆(x1 |x 2 |x 2 x 2 )∆(x 1 |x 2 |y)□ 1 x 1 x 2 := ∆(x1 x 2 |x 2 |y)∆(x 1 |x 2 |y)□ 1 x 2 y := ∆(x2 y|x 2 |y)∆(x 1 |x 2 |y)□ 2 x 1 x 2 := ∆(x1 |x 1 x 2 |y)∆(x 1 |x 2 |y)□ 2 x 2 y := ∆(x1 |x 2 y|y)∆(x 1 |x 2 |y)□ 3 x 1 x 2 := ∆(x1 |x 2 |x 1 x 2 )∆(x 1 |x 2 |y)□ 3 x 2 y := ∆(x1 |x 2 |x 2 y)∆(x 1 |x 2 |y)□ 1 x 1 y := ∆(x1 y|x 2 |y)∆(x 1 |x 2 |y)□ 1 yy := ∆(yy|x2 |y)∆(x 1 |x 2 |y)□ 2 x 1 y := ∆(x1 |x 1 y|y)∆(x 1 |x 2 |y)□ 2 yy := ∆(x1 |yy|y)∆(x 1 |x 2 |y)□ 3 x 1 y := ∆(x1 |x 2 |x 1 y)∆(x 1 |x 2 |y)□ 3 yy := ∆(x1 |x 2 |yy)∆(x 1 |x 2 |y) .Thanks to these notations, we can rewrite the three equations (2.10),(2.22) and (2.23) in a more compact style.Lemma 2.29. A completely integrable system of three second order partialdifferential equations(2.30)⎧⎪⎨y x 1 x 1(x) = F ( 1,1 x 1 , x 2 , y(x), y x 1(x), y x 2(x) ) ,y x⎪⎩1 x 2(x) = F ( 1,2 x 1 , x 2 , y(x), y x 1(x), y x 2(x) ) ,y x 2 x 2(x) = F ( 2,2 x 1 , x 2 , y(x), y x 1(x), y x 2(x) ) ,is equivalent to the simplest system Y X 1 X 1 = 0, Y X 1 X 2 = 0, Y X 2 ,X 2 = 0, ifand only if there exist local K-analytic functions X 1 , X 2 , Y such that it may

e written under the specific form:(2.31)⎧)y x 1 x 1 = −□3 x 1 x+ y 1 x 1 ·(−2□ 3 x 1 y + □1 x 1 x+ y 1 x 2 · (□ 2 )x 1 x +( ) ( )1+ y x 1 y x 1 · −□ 3 yy + 2□ 1 x 1 y+ y x 1 y x 2 · 2□ 2 x 1 y++ y x 1 y x 1 y x 1 · (□ 1 yy)+ yx 1 y x 1 y x 2 · (□ 2 yy),))y x⎪⎨1 x 2 = −□3 x 1 x+ y 2 x 1 ·(−□ 3 x 2 y + □1 x 1 x+ y 2 x 2 ·(−□ 3 x 1 y + □2 x 1 x+( ) ()2+ y x 1 y x 1 · □ 1 x 2 y+ y x 1 y x 2 · −□ 3 yy + □ 1 x 1 y + □2 x 2 y+( )+ y x 2 y x 2 · □ 2 x 1 y+ y x 1 y x 1 y x 2 · (□ 1 yy)+ yx 1 y x 2 y x 2 · (□ 2 yy),y x 2 x 2 = −□3 x 2 x+ y 2 x 1 · (□ 1 ) )x 2 x + 2 yx 2 ·(−2□ 3 x 2 y + □2 x 2 x+( ) (2 )+ y x 1 y x 2 · 2□ 1 x⎪⎩2 y+ y x 2 y x 2 · −□ 3 yy + 2□2 x 2 y++ y x 1 y x 2 y x 2 · (□ 1 yy)+ yx 2 y x 2 y x 2 · (□ 2 yy).3692.32. General formulas. The formal dictionary between the original determinantialformulas (2.10), (2.22), (2.23), between the coefficients (2.15) ofthe second order prolongation of a vector field and between the new squareformulas (2.31) above is evident. Consequently, without any computation,just by translating the family of formulas (2.14), we may deduce the exactformulation of the desired generalization of Lemma 2.29 above.Lemma 2.33. A completely integrable system of second order partial differentialequations of the form(2.34)y x j 1x j 2 (x) = F j 1,j 2(x, y(x), y x 1(x), . . .,y x n(x)) , j 1 , j 2 = 1, . . .n,is equivalent to the simplest system Y X j 1X j 2 = 0, j 1 , j 2 = 1, . . ., n, if andonly if there exist local K-analytic functions X l , Y such that it may be writtenunder the specific form:(2.35) ⎧⎪⎨⎪⎩y x j 1x j 2 = −□ n+1x j 1x j 2 + n∑+y x j 1 ·k 1 =1y x k 1 ·(□ k 1x j 2y − 1 )2 δk 1j 2□ n+1yy+y x j 1 y x j 2 · (□ k 1yy)}.{()□ k 1x j 1x − j 2 δk 1j 1□ n+1x j 2y − δk 1j 2□ n+1 +x j 1y+ y x j 2 ·(□ k 1x j 1y − 1 )2 δk 1j 1□ n+1yy +

370Of course, to define the square functions in the context of n 2 independentvariables (x 1 , x 2 , . . ., x n ), we introduce the Jacobian determinantX 1 x· · · X 1 ∣ 1 x n X1 y ∣∣∣∣∣∣∣ (2.36) ∆(x 1 |x 2 | · · · |x n |y) :=. · · · . .,X n x· · · X n∣1 x n Xn yY x 1 · · · Y x n Y ytogether with its modifications(2.37) ∆ ( x 1 | · · · | k 1x j 1x j 2| · · · |y ) ,in which the k 1 -th column of partial first order derivatives | k 1x k 1| is replacedby the column | k 1x j 1x j 2| of partial derivatives. Here, the indices k 1 , j 1 ,j 2 satisfy 1 k 1 , j 1 , j 2 n + 1, with the convention that we adopt thenotational equivalence(2.38) x n+1 ≡ y .This convention will be convenient to write some of our general formulas inthe sequel.As we promised to only summarize the proof of Theorem 1.7 in this paper,we will not develope the proof of Lemma 2.33: it is similar to the proof ofLemma 3.32 in [Me2004].§3. FIRST AND SECOND AUXILIARY SYSTEM3.1. Functions G j1 ,j 2, H k 1j 1 ,j 2, L k 1j 1and M k 1. To discover the four families offunctions appearing in the statement of Theorem 1.7, by comparing (2.35)and (1.10), it suffices (of course) to set:(3.2)⎧⎪⎨⎪⎩G j1 ,j 2:= −□ n+1x j 1x , j 2H k 1j 1 ,j 2:= □ k 1x j 1x − j 2 δk 1j 1□ n+1x j 2y − δk 1j 2□ n+1x j 1y ,L k 1j 1:= 2 □ k 1x j 1y − δk 1j 1□ n+1yy ,M k 1:= □ k 1yy.Consequently, we have shown the “only if” part of Theorem 1.7, which isthe easiest implication.To establish the “if” part, by far the most difficult implication, the verymain lemma can be stated as follows.Lemma 3.3. The partial differerential relations (I’), (II’), (III’) and (IV’)which express in length the compatibility conditions (1.11) are necessaryand sufficient for the existence of functions X l , Y of (x l 1, y) satisfying thesecond order nonlinear system of partial differential equations (3.2) above.

Indeed, the collection of equations (3.2) is a system of partial differentialequations with unknowns X l , Y , by virtue of the definition of the squarefunctions.3.4. First auxiliary system. To proceed further, we observe that there are(m + 1) more square functions than functions G j1 ,j 2, H k 1j 1 ,j 2, L k 1j 1and M k 1.Indeed, a simple counting yields:⎧#{□⎪⎨k 1x j 1x } = n2 (n + 1), #{□ k j 122x j 1y } = n2 ,(3.5)#{□ k 1yy } = n,n(n + 1)#{□n+1 } = ,x⎪⎩j 1x j 22#{□ n+1x j 1y } = n,#{□n+1 yy } = 1,whereas⎧⎨#{G j 1,j 2n(n + 1)} = , #{H k 1j(3.6)21 ,j 2} = n2 (n + 1),2⎩#{L k 1j 1} = n 2 , #{M k 1} = n.Here, the indices j 1 , j 2 , k 1 satisfy 1 j 1 , j 2 , k 1 n. Similarly asin [Me2004], to transform the system (3.2) in a true complete system, letus introduce functions Π k 1j 1 ,j 2of (x l 1, y), where 1 j 1 , j 2 , k 1 n+1, whichsatisfy the symmetry Π k 1j 1 ,j 2= Π k 1j 1 ,j 1, and let us introduce the following firstauxiliary system:{□k 1x(3.7)j 1x = j 2 Πk 1j 1 ,j 2, □ k 1x j 1y = Πk 1j 1 ,n+1 , □k 1yy = Πk 1n+1,n+1 ,□ n+1x j 1x = j 2 Πn+1 j 1 ,j 2, □ n+1x j 1y = Πn+1 j 1 ,n+1 , □n+1 yy = Π n+1n+1,n+1 .It is complete. The necessary and sufficient conditions for the existence ofsolutions X l , Y follow by cross differentiations.Lemma 3.8. For all j 1 , j 2 , j 3 , k 1 = 1, 2, . . ., n+1, we have the cross differentiationrelations(3.9)(□k 1)x j 1x −( )□ k j 12 x j 3 x j 1x j 3= − ∑n+1x j 2k 2 =1□ k 2x j 1x j 2 □k 1x j 3x + ∑n+1k 2k 2 =1371□ k 2x j 1x j 3 □k 1x j 2x k 2 .The proof of this lemma is exactly the same as the proof of Lemma 3.40in [Me2004].As a direct consequence, we deduce that a necessary and sufficient conditionfor the existence of solutions Π k 1j 1 ,j 2to the first auxiliary system is thatthey satisfy the following compatibility partial differential relations:(3.10)∂Π k 1j 1 ,j 2∂x j 3− ∂Πk 1j 1 ,j 3∂x j 2= −∑n=1k 2 =1∑n=1Π k 2j 1 ,j 2 · Π k 1j 3 ,k 2+k 2 =1Π k 2j 1 ,j 3 · Π k 1j 2 ,k 2,

372for all j 1 , j 2 , j 3 , k 1 = 1, . . .,n + 1.We shall have to specify this system in length according to the splitting{1, 2, . . ., n} and {n + 1} of the indices of coordinates. We obtain six familiesof equations equivalent to (3.10) just above:(3.11) ⎧⎪⎨⎪⎩( )Πn+1j 1 ,j − ( )2Π n+1x j 3 j 1 ,j = − 3 x j 2n∑k 2 =1+( )Πn+1j 1 ,j − ( n∑2Π n+1y j 1 ,n+1)x = − j 2k 2 =1(Πn+1j 1 ,n+1)y − ( n∑Π n+1n+1,n+1)x = − j 1(Πk 1j 1 ,j 2)x j 3 − ( Π k 1j 1 ,j 3)x j 2 = −+k 2 =1+n∑k 2 =1+(Πk 1)j 1 ,j − ( n∑2Π k 1y j 1 ,n+1)x = − j 2k 2 =1(Πk 1)j 1 ,n+1 − ( n∑Π k 1y n+1,n+1)x = − j 1+k 2 =1+Π k 2j 1 ,j 2Π n+1j 3 ,k 2− Π n+1j 1 ,j 2Π n+1j 3 ,n+1 +n∑k 2 =1Π k 2j 1 ,j 3Π n+1j 2 ,k 2+ Π n+1j 1 ,j 3Π n+1j 2 ,n+1 ,Π k 2j 1 ,j 2Π n+1n+1,k 2− Π n+1j 1 ,j 2Π n+1n+1,n+1 +n∑k 2 =1Π k 2j 1 ,n+1 Πn+1 j 2 ,k 2+ Π n+1j 1 ,n+1 Πn+1 j 2 ,n+1 ,Π k 2j 1 ,n+1 Πn+1 n+1,k 2− Π n+1j 1 ,n+1 Πn+1 n+1,n+1 +a ◦n∑k 2 =1Π k 2n+1,n+1 Π n+1j 1 ,k 2+ Π n+1n+1,n+1 Π n+1j 1 ,n+1 ,a ◦Π k 2j 1 ,j 2Π k 1j 3 ,k 2− Π n+1j 1 ,j 2Π k 1j 3 ,n+1 +n∑k 2 =1Π k 2j 1 ,j 3Π k 1j 2 ,k 2+ Π n+1j 1 ,j 3Π k 1j 2 ,n+1 ,Π k 2j 1 ,j 2Π k 1n+1,k 2− Π n+1j 1 ,j 2Π k 1n+1,n+1 +n∑k 2 =1Π k 2j 1 ,n+1 Πk 1j 2 ,k 2+ Π n+1j 1 ,n+1 Πk 1j 2 ,n+1 ,Π k 2j 1 ,n+1 Πk 1n+1,k 2− Π n+1j 1 ,n+1 Πk 1n+1,n+1 +n∑k 2 =1Π k 2n+1,n+1 Π k 1j 1 ,k 2+ Π n+1n+1,n+1 Π k 1j 1 ,n+1 .where the indices j 1 , j 2 , j 3 , k 1 vary in the set {1, 2, 1, . . ., n}.

3.12. Principal unknowns. As there are (m + 1) more square (or Pi) functionsthan the functions G j1 ,j 2, H k 1j 1 ,j 2, L k 1j 1and M k 1, we cannot invert directlythe linear system (3.2). To quasi-inverse it, we choose the (m + 1) specificsquare functions(3.13) Θ 1 := □ 1 x 1 x 1, Θ2 := □ 2 x 2 x 2, · · · · · · , Θn+1 := □ n+1x n+1 x n+1 ,calling them principal unknowns, and we get the quasi-inversion:(3.14) ⎧Π k 1j 1 ,j 2= □ k 1x j 1x = j 2 Hk 1j 1 ,j 2− 1 2 δk 1j 1H j 2j 2 ,j 2− 1 2 δk 1j 2H j 1j 1 ,j 1+ 1 2 δk 1j 1Θ j 2+ 1 2 δk 1j 2Θ j 1,⎪⎨⎪⎩Π k 1j 1 ,n+1 = □k 1x j 1y = 1 2 Lk 1j 1+ 1 2 δk 1j 1Θ n+1 ,Π k 1n+1,n+1 = □k 1yy = Mk 1,Π n+1j 1 ,j 2= □ n+1x j 1x j 2 = −G j 1 ,j 2,Π n+1j 1 ,n+1 = □n+1 x j 1y = −1 2 Hj 1j 1 ,j 1+ 1 2 Θj 1.3.15. Second auxiliary system. Replacing the five families of functionsΠ k 1j 1 ,j 2, Π k 1j 1 ,n+1 , Πk 1n+1,n+1 , Πn+1 j 1 ,j 2, Π n+1j 1 ,n+1 by their values obtained in (3.14)just above together with the principal unknowns(3.16) Π j 1j1 ,j 1= Θ j 1, Π n+1n+1,n+1 = Θn+1 ,in the six equations (3.11) 1 , (3.11) 2 , (3.11) 3 , (3.11) 4 , (3.11) 5 and (3.11) 6 ,after hard computations that we will not reproduce here, we obtain six familiesof equations. From now on, we abbreviate every sum ∑ nk=1 as ∑ k 1.Firstly:(3.17) 0 = G j1 ,j 2 ,x j 3 − G j1 ,j 3 ,x j 2 + ∑ k 1G j3 ,k 1H k 1j 1 ,j 2− ∑ k 1G j2 ,k 1H k 1j 1 ,j 3.This is (I’) of Theorem 1.7. Just above and below, we plainly underline themonomials involving a first order derivative. Secondly:(3.18) ⎧Θ j 1= −2 G x j 2 j 1 ,j 2 ,y + H j 1+ j 1 ,j 1 ,x j 2373⎪⎨+ ∑ k 1G j2 ,k 1L k 1j 1+ 1 2 Hj 1j 1 ,j 1H j 2j 2 ,j 2− ∑ k 1H k 1j 1 ,j 2H k 1k 1 ,k 1−− G j1 ,j 2Θ n+1 − 1 2 Hj 1j 1 ,j 1Θ j 2− 1 2 Hj 2j 2 ,j 2Θ j 1+ ∑ k 1H k 1j 1 ,j 2Θ k 1+⎪⎩+ 1 2 Θj 1Θ j 2.

374Thirdly:(3.19) ⎧−Θ n+1 + 1 x j 12 Θj 1y = 1 2 Hj 1j 1 ,j 1 ,y −⎪⎨− ∑ G j1 ,k 1M k 1+ 1 4k 1⎪⎩+ 1 4 Hj 1j 1 ,j 1Θ n+1 − 1 4∑∑Fourtly:(3.20) ⎧12 δk 1j 1Θ j 2− 1x j 32 δk 1j 1Θ j 3+ 1x j 22 δk 1j 2Θ j 1− 1x j 32 δk 1j 3Θ j 1= x j 2H k 1k 1 ,k 1L k 1j 1+k 1L k 1j 1Θ k 1− 1 4 Θj 1Θ n+1 .k 1= −H k 1j 1 ,j 2 ,x j 3 + Hk 1j 1 ,j 3 ,x j 2 − 1 2 δk 1j 1H j 3j 3 ,j 3 ,x j 2 + 1 2 δk 1j 1H j 2j 2 ,j 2 ,x j 3 −− 1 2 δk 1j 3H j 1j 1 ,j 1 ,x j 2 + 1 2 δk 1j 2H j 1j 1 ,j 1 ,x j 3 −⎪⎨− 1 2 G j 1 ,j 2L k 1j 3+ 1 2 G j 1 ,j 3L k 1j 2− 1 4 δk 1j 3H j 1j 1 ,j 1H j 2j 2 ,j 2+ 1 4 δk 1j 2H j 1j 1 ,j 1H j 3j 3 ,j 3−− ∑ k 2H k 2j 1 ,j 2H k 1j 3 ,k 2+ ∑ k 2H k 2j 1 ,j 3H k 1j 2 ,k 2− 1 2 δk 1j 2H k 2j 1 ,j 3H k 2k 2 ,k 2+ 1 2 δk 1j 3H k 2j 1 ,j 2H k 2k 2 ,k 2−− 1 2 δk 1j 2G j1 ,j 3Θ n+1 + 1 2 δk 1j 3G j1 ,j 2Θ n+1 −⎪⎩− 1 4 δk 1j 2H j 1j 1 ,j 1Θ j 3+ 1 4 δk 1j 3H j 1j 1 ,j 1Θ j 2− 1 4 δk 1j 2H j 3j 3 ,j 3Θ j 1+ 1 4 δk 1j 3H j 2j 2 ,j 2Θ j 1−− 1 ∑2 δk 1j 3 j 1 ,j 2Θ k 2+ 1 ∑2 δk 1j 2 j 1 ,j 3Θ k 2−k 1H k 2− 1 4 δk 1j 3Θ j 1Θ j 2+ 1 4 δk 1j 2Θ j 1Θ j 3.k 1H k 2Fifthly:(3.21)⎧12 δk 1j 1Θ j 2y + 1 2 δk 1j 2Θ j 1y − 1 2 δk 1j 1Θ n+1 =x j 2⎪⎨= G j1 ,j 2M k 1+ 1 ∑H k 1j2 1 ,k 2L k 2j 1− 1 ∑H k 2j2 1 ,j 2L k 1k 2− 1 ∑4 δk 1j 2k 2 k 2⎪⎩ − 1 4 δk 1j 2H j 1j 1 ,j 1Θ n+1 + 1 ∑4 δk 1j 2L k 2j 1Θ k 2+ 1 4 δk 1j 2Θ j 1Θ n+1 .k 2H k 2k 2 ,k 2L k 2j 1−k 2

375Sixthly:(3.22) ⎧δ k 1j 1Θ n+1y = −L k 1j 1 ,y + 2 Mk 1+ x j 1⎪⎨ + 2 ∑ ∑H k 1j 1 ,k 2M k 2− δ k 1j 1k 2⎪⎩k 2H k 2k 2 ,k 2M k 2− 1 2∑+ δ k 1j 1M k 2Θ k 2+ 1 2 δk 1j 1Θ n+1 Θ n+1 .k 2∑L k 2j 1L k 1k 2+k 23.23. Solving Θ j 1, x j 2 Θj 1 y , Θ n+1 and Θ n+1x j 1 y . From the six families of equations(3.17), (3.18), (3.19), (3.20), (3.21) and (3.22), we can solve Θ j 1, x j 2Θ j 1 y , Θ n+1 and Θ n+1x j 1 y . Not mentioning the (hard) intermediate computations,we obtain firstly:(3.24)⎧⎪⎨Θ j 1x j 2 = −2 G j 1 ,j 2 ,y + H j 1j 1 ,j 1 ,x j 2 + ∑ lG j2 ,l L l j 1+ 1 2 Hj 1j 1 ,j 1H j 2j 2 ,j 2− ∑ lH l j 1 ,j 2H l l,l−⎪⎩− G j1 ,j 2Θ n+1 − 1 2 H j 1 ,j 1Θ j 1− 1 2 Hj 2j 2 ,j 2Θ j 1+ ∑ lH l j 1 ,j 2Θ l + 1 2 Θj 1Θ j 2.Secondly:(3.25) ⎧Θ j 1y = − 1 3 Hj 1j 1 ,j 1 ,y + 2 3 Lj 1+ 4j 1 ,x j 13 G j 1 ,j 1M j 1+ 2 3⎪⎨⎪⎩+ 2 3∑l+ 1 2 Θj 1Θ n+1 .H j 1j 1 ,l Ll j 1− 2 3∑l∑G j1 ,l M l − 1 ∑Hl,l l L l j21+ll∑L l j 1Θ l +H l j 1 ,j 1L j 1l− 1 2 Hj 1j 1 ,j 1Θ n+1 + 1 2lThirdly:(3.26) ⎧Θ n+1 = − 2 x j 13 Hj 1j 1 ,j 1 ,y + 1 3 Lj 1+ 2j 1 ,x j 13 G j 1 ,j 1M j 1+ 4 3⎪⎨⎪⎩+ 1 3∑l+ 1 2 Θj 1Θ n+1 .H j 1j 1 ,l Ll j 1− 1 3∑l∑G j1 ,l M l − 1 ∑Hl,l l 2Ll j 1+ll∑L l j 1Θ l +H l j 1 ,j 1L j 1l− 1 2 Hj 1j 1 ,j 1Θ n+1 + 1 2l

376Fourtly:(3.27) ⎧Θ ⎪⎨n+1y = −L j 1j1 ,y + 2 Mj 1+ 2 ∑x j 1lH j 1j 1 ,l Ml − ∑ lH l l,l M l − 1 2∑lL l j 1L j 1l+⎪⎩+ ∑ lM l Θ l + 1 2 Θn+1 Θ n+1 .These four families of partial differential equations constitute the secondauxiliary system. By replacing these solutions in the three remaining familiesof equations (3.20), (3.21) and (3.22), we obtain supplementary equations(which we do not copy) that are direct consequences of (I’), (II’), (III’),(IV’).To complete the proof of the main Lemma 3.3 above, it suffices now toestablish the first implication of the following list, since the other three havebeen already established.• Some given functions G j1 ,j 2, H k 1j 1 ,j 2, L k 1j 1and M k 1of (x l 1, y) satisfy thefour families of partial differential equations (I’), (II’), (III’) and (IV’)of Theorem 1.7.⇓• There exist functions Θ j 1, Θ n+1 satisfying the second auxiliary system(3.24), (3.25), (3.26) and (3.27).⇓• These solution functions Θ j 1, Θ n+1 satisfy the six families of partialdifferential equations (3.17), (3.18), (3.19), (3.20), (3.21) and (3.22).⇓• There exist functions Π k 1j 1 ,j 2of (x l 1, y), 1 j 1 , j 2 , k 1 m + 1, satisfyingthe first auxiliary system (3.7) of partial differential equations.⇓• There exist functions X l 2, Y of (x l 1, y) transforming the systemy x j 1x j 2 = F j 1,j 2(x l 1, y, y x l 2 ), j 1 , j 2 = 1, . . ., n, to the simplest systemY X j 1X j 2 = 0, j 1 , j 1 = 1, . . ., n.3.28. Compatibility conditions for the second auxiliary system. We noticethat the second auxiliary system is also a complete system. Thus, toestablish the first above implication, it suffices to show that the four familiesof compatibility conditions:⎧0 = ( Θ j )1− ( Θ j )1,x j 2 x⎪⎨j 3 x j 3 x j 20 = ( Θ j )1x(3.29)− ( Θ j j 12 y y)x , j 20 = ( Θ n+1 )x − ( )Θ n+1j 2⎪⎩x j 10 = ( Θ n+1x j 2x j 2)y − ( )Θ n+1yx j 1 ,x j 2 ,

are a consequence of (I’), (I”), (III’), (IV’).For instance, in (3.29) 1 , replacing Θ j 1by its expression (3.24), differentiatingit with respect to x j 3x j 2, replacing Θ j 1by its expression (3.24), differentiatingit with respect to x j 2x j 3and substracting, we get:(3.30) ⎧0 = −2 G j1 ,j 2 ,yx j 3 + 2 G j1 ,j 3 ,yx j 2 + H j 1j 1 ,j 1− H j 1,x j 2x j 3 j a 1 ,j 1+,x j 3x j 2 a+ 1 2 Θj 1x j 3 Θj 2+ 1 2 Θj 1Θ j 2x j 3 − 1 2 Θj 1x j 2 Θj 3− 1 2 Θj 1Θ j 3x j 2 −377− 1 2 Hj 1j 1 ,j 1 ,x j 3 Θj 2− 1 2 Hj 1j 1 ,j 1Θ j 2x j 3 + 1 2 Hj 1j 1 ,j 1 ,x j 2 Θj 3+ 1 2 Hj 1j 1 ,j 1Θ j 3x j 2 −⎪⎨− 1 2 Hj 2j 2 ,j 2 ,x j 3 Θj 1− 1 2 Hj 2j 2 ,j 2Θ j 1x j 3 + 1 2 Hj 3j 3 ,j 3 ,x j 2 Θj 1+ 1 2 Hj 3j 3 ,j 3Θ j 1x j 2 −− G j1 ,j 2 ,x j 3 Θ n+1 − G j1 ,j 2Θ n+1x j 3+ G j1 ,j 3 ,x j 2 Θ n+1 + G j1 ,j 3Θ n+1x j 2 ++ ∑ lH l j 1 ,j 2 ,x j 3Θ l + ∑ lH l j 1 ,j 2Θ l x j 3− ∑ lH l j 1 ,j 3 ,x j 2Θ l − ∑ lH l j 1 ,j 3Θ l x j 2++ 1 2 Hj 1j 1 ,j 1 ,x j 3 Hj 2j 2 ,j 2+ 1 2 Hj 1j 1 ,j 1H j 2j 2 ,j 2 ,x j 3 − 1 2 Hj 1j 1 ,j 1 ,x j 2 Hj 3j 3 ,j 3− 1 2 Hj 1j 1 ,j 1H j 3j 3 ,j 3 ,x j 2 −− ∑ lH l j 1 ,j 2 ,x j 3H l l,l − ∑ lH l j 1 ,j 2H l l,l,x j 3+ ∑ lH l j 1 ,j 3 ,x j 2H l l,l + ∑ lH l j 1 ,j 3H l l,l,x j 2+⎪⎩+ ∑ lG j2 ,l,x j 3 L l j 1+ ∑ lG j2 ,l L l j 1 ,x j 3− ∑ lG j3 ,l,x j 2 L l j 1− ∑ lG j3 ,l L l j 1 ,x j 2.Next, replacing the twelve first order partial derivatives underlined justabove:(3.31)⎧⎨⎩Θ j 1x j 3 , Θj 2x j 3 , Θj 1x j 2 , Θj 3x j 2 , Θj 2x j 3 , Θj 3x j 2 ,Θ j 1x j 3 , Θj 1x j 2 ,Θn+1 x j 3 ,Θn+1 x j 2 , Θl x j 3, Θ l x j 2.by their values issued from (3.24), (3.26) and adapting the summation indices,we get the explicit developed form of the first family of compatibility

378conditions (3.29) 1 :(3.32) ⎧0 =? = −2 G j1 ,j 2 ,x j 3y + 2 G j1 ,j 3 ,x j 2y−− ∑ lG j3 ,l,x j 2 L l j 1+ ∑ lG j2 ,l,x j 3 L l j 1− G j1 ,j 2 ,y H j 3j 3 ,j 3+ G j1 ,j 3 ,y H j 2j 2 ,j 2−− 2 ∑ lG l,j3 H l j 1 ,j 2+ 2 ∑ lG l,j2 H l j 1 ,j 3− ∑ lH l j 1 ,j 2 ,x j 3H l l,l + ∑ lH l j 1 ,j 3 ,x j 2H l l,l −− 2 3 Hj 2j 2 ,j 2 ,y G j 1 ,j 3+ 2 3 Hj 3j 3 ,j 3 ,y G j 1 ,j 2− 2 3 Lj 3j 3 ,x j 3 G j 1 ,j 2+ 2 3 Lj 2j 2 ,x j 2 G j 1 ,j 3−− ∑ lL l j 1 ,x j 2G j3 ,l + ∑ lL l j 1 ,x j 3G j2 ,l−⎪⎨− 2 3 G j 1 ,j 2G j3 ,j 3M j 3+ 2 3 G j 1 ,j 3G j2 ,j 2M j 2− 4 3+ 4 ∑G j1 ,j33G j2 ,l M l − 1 ∑2ll− 1 ∑G j3 ,l H j 2j22 ,j 2L l j 1+ 1 ∑2ll+ 1 ∑G j1 ,j22Hl,l l Ll j 3− 1 ∑3llG j3 ,l H j 1j 1 ,j 1L l j 2+ 1 2G j2 ,l H j 3j 3 ,j 3L l j 1− 1 2G j1 ,j 2H j 3j 3 ,l Ll j 3+ 1 3∑G j1 ,j 2G j3 ,l M l +l∑lG j2 ,l H j 1j 1 ,j 1L l j 3−∑G j1 ,j 3Hl,l l Ll j 2+l∑lG j1 ,j 3H j 2j 2 ,l Ll j 2−⎪⎩− 1 3 G j 1 ,j 3Hj l 2 ,j 2L j 2l+ 1 3 G j 1 ,j 2Hj l 3 ,j 3L j 3l−− ∑ ∑G j2 ,p Hj l 1 ,j 3L p l + ∑ ∑G j3 ,p Hj l 1 ,j 2L p l −l pl p− ∑ ∑Hj l 1 ,j 2H p l,j 3Hp,p p + ∑ ∑Hj l 1 ,j 3H p l,j 2Hp,p.pl pl pLemma 3.33. ([Me2003, Me2004]) This first family of compatibility conditionsfor the second auxiliary system obtained by developing (3.29) 1 inlength, together with the three remaining families obtained by developing(3.29) 2 , (3.29) 3 , (3.29) 4 in length, are consequences, by linear combinationsand by differentiations, of (I’), (II’), (III’), (IV’), of Theorem 1.7.The summarized proof of Theorem 1.7 is complete.

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