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

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