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

378conditions (3.29) 1 :(3.32) ⎧0 =? = −2 G j1 ,j 2 ,x j 3y + 2 G j1 ,j 3 ,x j 2y−− ∑ lG j3 ,l,x j 2 L l j 1+ ∑ lG j2 ,l,x j 3 L l j 1− G j1 ,j 2 ,y H j 3j 3 ,j 3+ G j1 ,j 3 ,y H j 2j 2 ,j 2−− 2 ∑ lG l,j3 H l j 1 ,j 2+ 2 ∑ lG l,j2 H l j 1 ,j 3− ∑ lH l j 1 ,j 2 ,x j 3H l l,l + ∑ lH l j 1 ,j 3 ,x j 2H l l,l −− 2 3 Hj 2j 2 ,j 2 ,y G j 1 ,j 3+ 2 3 Hj 3j 3 ,j 3 ,y G j 1 ,j 2− 2 3 Lj 3j 3 ,x j 3 G j 1 ,j 2+ 2 3 Lj 2j 2 ,x j 2 G j 1 ,j 3−− ∑ lL l j 1 ,x j 2G j3 ,l + ∑ lL l j 1 ,x j 3G j2 ,l−⎪⎨− 2 3 G j 1 ,j 2G j3 ,j 3M j 3+ 2 3 G j 1 ,j 3G j2 ,j 2M j 2− 4 3+ 4 ∑G j1 ,j33G j2 ,l M l − 1 ∑2ll− 1 ∑G j3 ,l H j 2j22 ,j 2L l j 1+ 1 ∑2ll+ 1 ∑G j1 ,j22Hl,l l Ll j 3− 1 ∑3llG j3 ,l H j 1j 1 ,j 1L l j 2+ 1 2G j2 ,l H j 3j 3 ,j 3L l j 1− 1 2G j1 ,j 2H j 3j 3 ,l Ll j 3+ 1 3∑G j1 ,j 2G j3 ,l M l +l∑lG j2 ,l H j 1j 1 ,j 1L l j 3−∑G j1 ,j 3Hl,l l Ll j 2+l∑lG j1 ,j 3H j 2j 2 ,l Ll j 2−⎪⎩− 1 3 G j 1 ,j 3Hj l 2 ,j 2L j 2l+ 1 3 G j 1 ,j 2Hj l 3 ,j 3L j 3l−− ∑ ∑G j2 ,p Hj l 1 ,j 3L p l + ∑ ∑G j3 ,p Hj l 1 ,j 2L p l −l pl p− ∑ ∑Hj l 1 ,j 2H p l,j 3Hp,p p + ∑ ∑Hj l 1 ,j 3H p l,j 2Hp,p.pl pl pLemma 3.33. ([Me2003, Me2004]) This first family of compatibility conditionsfor the second auxiliary system obtained by developing (3.29) 1 inlength, together with the three remaining families obtained by developing(3.29) 2 , (3.29) 3 , (3.29) 4 in length, are consequences, by linear combinationsand by differentiations, of (I’), (II’), (III’), (IV’), of Theorem 1.7.The summarized proof of Theorem 1.7 is complete.