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

e written under the specific form:(2.31)⎧)y x 1 x 1 = −□3 x 1 x+ y 1 x 1 ·(−2□ 3 x 1 y + □1 x 1 x+ y 1 x 2 · (□ 2 )x 1 x +( ) ( )1+ y x 1 y x 1 · −□ 3 yy + 2□ 1 x 1 y+ y x 1 y x 2 · 2□ 2 x 1 y++ y x 1 y x 1 y x 1 · (□ 1 yy)+ yx 1 y x 1 y x 2 · (□ 2 yy),))y x⎪⎨1 x 2 = −□3 x 1 x+ y 2 x 1 ·(−□ 3 x 2 y + □1 x 1 x+ y 2 x 2 ·(−□ 3 x 1 y + □2 x 1 x+( ) ()2+ y x 1 y x 1 · □ 1 x 2 y+ y x 1 y x 2 · −□ 3 yy + □ 1 x 1 y + □2 x 2 y+( )+ y x 2 y x 2 · □ 2 x 1 y+ y x 1 y x 1 y x 2 · (□ 1 yy)+ yx 1 y x 2 y x 2 · (□ 2 yy),y x 2 x 2 = −□3 x 2 x+ y 2 x 1 · (□ 1 ) )x 2 x + 2 yx 2 ·(−2□ 3 x 2 y + □2 x 2 x+( ) (2 )+ y x 1 y x 2 · 2□ 1 x⎪⎩2 y+ y x 2 y x 2 · −□ 3 yy + 2□2 x 2 y++ y x 1 y x 2 y x 2 · (□ 1 yy)+ yx 2 y x 2 y x 2 · (□ 2 yy).3692.32. General formulas. The formal dictionary between the original determinantialformulas (2.10), (2.22), (2.23), between the coefficients (2.15) ofthe second order prolongation of a vector field and between the new squareformulas (2.31) above is evident. Consequently, without any computation,just by translating the family of formulas (2.14), we may deduce the exactformulation of the desired generalization of Lemma 2.29 above.Lemma 2.33. A completely integrable system of second order partial differentialequations of the form(2.34)y x j 1x j 2 (x) = F j 1,j 2(x, y(x), y x 1(x), . . .,y x n(x)) , j 1 , j 2 = 1, . . .n,is equivalent to the simplest system Y X j 1X j 2 = 0, j 1 , j 2 = 1, . . ., n, if andonly if there exist local K-analytic functions X l , Y such that it may be writtenunder the specific form:(2.35) ⎧⎪⎨⎪⎩y x j 1x j 2 = −□ n+1x j 1x j 2 + n∑+y x j 1 ·k 1 =1y x k 1 ·(□ k 1x j 2y − 1 )2 δk 1j 2□ n+1yy+y x j 1 y x j 2 · (□ k 1yy)}.{()□ k 1x j 1x − j 2 δk 1j 1□ n+1x j 2y − δk 1j 2□ n+1 +x j 1y+ y x j 2 ·(□ k 1x j 1y − 1 )2 δk 1j 1□ n+1yy +