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

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.