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

372for all j 1 , j 2 , j 3 , k 1 = 1, . . .,n + 1.We shall have to specify this system in length according to the splitting{1, 2, . . ., n} and {n + 1} of the indices of coordinates. We obtain six familiesof equations equivalent to (3.10) just above:(3.11) ⎧⎪⎨⎪⎩( )Πn+1j 1 ,j − ( )2Π n+1x j 3 j 1 ,j = − 3 x j 2n∑k 2 =1+( )Πn+1j 1 ,j − ( n∑2Π n+1y j 1 ,n+1)x = − j 2k 2 =1(Πn+1j 1 ,n+1)y − ( n∑Π n+1n+1,n+1)x = − j 1(Πk 1j 1 ,j 2)x j 3 − ( Π k 1j 1 ,j 3)x j 2 = −+k 2 =1+n∑k 2 =1+(Πk 1)j 1 ,j − ( n∑2Π k 1y j 1 ,n+1)x = − j 2k 2 =1(Πk 1)j 1 ,n+1 − ( n∑Π k 1y n+1,n+1)x = − j 1+k 2 =1+Π k 2j 1 ,j 2Π n+1j 3 ,k 2− Π n+1j 1 ,j 2Π n+1j 3 ,n+1 +n∑k 2 =1Π k 2j 1 ,j 3Π n+1j 2 ,k 2+ Π n+1j 1 ,j 3Π n+1j 2 ,n+1 ,Π k 2j 1 ,j 2Π n+1n+1,k 2− Π n+1j 1 ,j 2Π n+1n+1,n+1 +n∑k 2 =1Π k 2j 1 ,n+1 Πn+1 j 2 ,k 2+ Π n+1j 1 ,n+1 Πn+1 j 2 ,n+1 ,Π k 2j 1 ,n+1 Πn+1 n+1,k 2− Π n+1j 1 ,n+1 Πn+1 n+1,n+1 +a ◦n∑k 2 =1Π k 2n+1,n+1 Π n+1j 1 ,k 2+ Π n+1n+1,n+1 Π n+1j 1 ,n+1 ,a ◦Π k 2j 1 ,j 2Π k 1j 3 ,k 2− Π n+1j 1 ,j 2Π k 1j 3 ,n+1 +n∑k 2 =1Π k 2j 1 ,j 3Π k 1j 2 ,k 2+ Π n+1j 1 ,j 3Π k 1j 2 ,n+1 ,Π k 2j 1 ,j 2Π k 1n+1,k 2− Π n+1j 1 ,j 2Π k 1n+1,n+1 +n∑k 2 =1Π k 2j 1 ,n+1 Πk 1j 2 ,k 2+ Π n+1j 1 ,n+1 Πk 1j 2 ,n+1 ,Π k 2j 1 ,n+1 Πk 1n+1,k 2− Π n+1j 1 ,n+1 Πk 1n+1,n+1 +n∑k 2 =1Π k 2n+1,n+1 Π k 1j 1 ,k 2+ Π n+1n+1,n+1 Π k 1j 1 ,n+1 .where the indices j 1 , j 2 , j 3 , k 1 vary in the set {1, 2, 1, . . ., n}.