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

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.