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

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