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

7where the indices j, l 1 vary in {1, 2, . . ., m};⎧0 = − 1 2 Hj + 1 l 1 ,y l 26 δj l 1H l 2+ 1l 2 ,y l 23 δj l 2H l 1+ l 1 ,y l 1+ L j l 1 ,l 2 ,x − 1 3 δj l 1L l 2l2 ,l 2 ,x − 2 3 δj l 2L l 1l1 ,l 1 ,x +(II)⎪⎨⎪⎩+ G j M l1 ,l 2− 1 3 δj l 1G l 2M l2 ,l 2− 2 3 δj l 2G l 1M l1 ,l 1++ 1 ∑ m3 δj l 1G k M l2 ,k − 1 3 δj l 2− 1 2k=1m∑H j k Lk l 1 ,l 2+ 1 2k=1(+ δ j 1l 16(+ δ j 1l 23m∑k=1m∑k=1m∑k=1m∑Hl k 1L j l 2 ,k +k=1H l 2kL k l 2 ,l 2− 1 6H l 1kL k l 1 ,l 1− 1 3m∑k=1m∑k=1G k M l1 ,k−H k l 2L l 2l2 ,kH k l 1L l 1l1 ,kwhere the indices j, l 1 , l 2 vary in {1, 2, . . ., m};⎧0 = L j − l 1 ,l 2 ,y l 3 Lj + l 1 ,l 3 ,y l 2 δj l 3M l1 ,l 2 ,x − δ j l 2M l1 ,l 3 ,x+))+,(III)(IV)⎪⎨⎪⎩+ 1 2 Hj l 3M l1 ,l 2− 1 2 Hj l 2M l1 ,l 3++ 1 ∑ m2 δj l 1Hl k 3M l2 ,k − 1 2 δj l 1k=1+ 1 ∑ m2 δj l 3Hl k 1M l2 ,k − 1 2 δj l 2+k=1m∑k=1∑ mk=1m∑mL k l 1 ,l 3L j l 2 ,k − ∑L k l 1 ,l 2L j l 3 ,k ,k=1k=1H k l 2M l3 ,k+H k l 1M l3 ,k+where the indices j, l 1 , l 2 , l 3 vary in {1, . . .m}; and{m∑m∑0 = M l1 ,l 2 ,y l 3 − M l1 ,l 3 ,y l 2 − L k l 1 ,l 2M l3 ,k + L k l 1 ,l 3M l2 ,k,k=1where the indices l 1 , l 2 , l 3 vary in {1, . . ., m}.Let us provide commentaries and explanations. The form of the righthandside of (3)(i) is the analog of the form of the right-hand side F in (4)of Lie’s Theorem 1.2. However, we notice that the right-hand side of (3)(i)k=1