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

58⇓• The functions G j , H j l 1, L j l 1 ,l 2and M l1 ,l 2satisfy the four families ofpartial differential equations (I), (II), (III) and (IV) of Theorem 1.7 (3).The four families of first order partial differential equations (3.99),(3.100), (3.101) and (3.102) satisfied by the principal unknowns will becalled the second auxiliary system. It is a complete system.To achieve the proof of Theorem 1.7 (3), we have to establish the reverseimplications. More precisely:• Some given functions G j , H j l 1, L j l 1 ,l 2= L j l 2 ,l 1and M l1 ,l 2= M l2 ,l 1of(x, y) satisfy the four families of partial differential equations (I), (II),(III) and (IV) of Theorem 1.7 (3), or equivalently, the partial differentialequations (3.106), (3.108), (3.110) and (3.96).⇓• There exist functions Θ 0 , Θ j satisfying the second auxiliary system(3.99), (3.100), (3.101) and (3.102).⇓• These solution functions Θ 0 , Θ j satisfy the six families of partial differentialequations (3.69), (3.86), (3.89), (3.91), (3.93) and (3.96).⇓• There exist functions Π j l 1 ,l 2of (x, y), 0 j, l 1 , l 2 m, satisfying thefirst auxiliary system (3.38) of partial differential equations.⇓• There exist functions X, Y j of (x, y) transforming the system y j xx =F j (x, y, y x ), j = 1, . . ., m, to the free particle system Y j XX = 0, j =1, . . ., m.The above last three implications have been already implicitely establishedin the preceding paragraphs, as may be checked by inspectingLemma 3.40 and the formal computations after §3.62.Thus, it remains only to establish the first implication in the above reverselist. Since the second auxiliary system (3.99), (3.100), (3.101) and (3.102)is complete and of first order, a necessary and sufficient condition for theexistence of solutions follows by writing out the following four families ofcross-differentiations:⎧ ( ))y − l 1(3.112)0 = ( Θ 0 x( ) (⎪⎨ 0 = Θ 0 y l 1−y l 20 = ( ) (Θ l 1− x y l 2( )⎪⎩ 0 = Θ l 1−y l 2y l 3Θ 0 y l 1Θ 0 y l 2Θ l 1y l 2,)x(Θ l 1y l 3),x)y l 1y l 2,.