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

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.