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

Open problem 1.17. For n = 2 establish a complete list of normal forms ofall possible systems (1.?) according to their Lie symmetry group. In case ofsuccess, classify Levi nondegenerate real analytic hypersurfaces of C 3 up tobiholomorphisms.361§2. COMPLETELY INTEGRABLE SYSTEMS OFSECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS2.1. Prolongation of a point transformation to the second order jetspace. Let K = R or C, let n ∈ N, suppose n 2, let x = (x 1 , . . .,x n ) ∈K n and let y ∈ K. According to the main assumption of Theorem 1.7, wehave to consider a local K-analytic diffeomorphism of the form((2.2) xj 1, y ) ↦−→ ( X j (x j 1, y), Y (x j 1, y) ) ,which transforms the system (1.8) to the system Y X i 1X i 2 = 0, 1 j 1 , j 2 n. Without loss of generality, we shall assume that this transformation isclose to the identity. To obtain the precise expression (2.35) of the transformedsystem (1.8), we have to prolong the above diffeomorphism to thesecond order jet space. We introduce the coordinates (x j , y, y x j 1 , y x j 1x j 2 ) onthe second order jet space. Let(2.3) D k := ∂∂x k + y x k∂n∑∂y +l=1y x k x l∂,∂y x lbe the k-th total differentiation operator. According to [Ol1986, BK1989,Ol1995], for the first order partial derivatives, one has the (implicit, compact)expression:⎛ ⎞ ⎛D 1 X 1 · · · D 1 X n −1 ⎛ ⎞D 1 Y(2.4)⎝Y X 1.Y X n⎠ = ⎝⎞. · · · .⎠D n X 1 · · · D n X n⎝.D n Ywhere (·) −1 denotes the inverse matrix, which exists, since the transformation(2.2) is close to the identity. For the second order partial derivatives,again according to [Ol1986, BK1989, Ol1995], one has the (implicit, compact)expressions:⎛ ⎞ ⎛Y X j X 1 D 1 X 1 · · · D 1 X n ⎞−1 ⎛ ⎞D 1 Y X j(2.5) ⎝.⎠ = ⎝. · · · .⎠ ⎝.⎠ ,Y X j X n D n X 1 · · · D n X n D n Y X jfor j = 1, . . ., n. Let DX denote the matrix (D i X j ) 1jn1in, where i is theindex of lines and j the index of columns, let Y X denote the column matrix(Y X i) 1inand let DY be the column matrix (D i Y ) 1in.⎠ ,