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

where(7.33) ∆ := ∑1kn∂ 2∂x k ∂a k .Exercise: solving (a k , b) from y = Π and y x l = Π x l, with Π as above,deduce a complete normal form for (E 2 ).Open problem 7.34. Find complete normal forms for submanifolds of solutionsassociated to (E 4 ) and to (E 5 ).§8. STUDY OF TWO SPECIFIC EXAMPLES8.1. Study of the Lie symmetries of (E 4 ). Its submanifold of solutions possessestwo equations:(8.2) y 1 = Π 1 (x, a, b 1 , b 2 ) y 2 = Π 2 (x, a, b 1 , b 2 ).For instance, a generic submanifold M ⊂ C 3 of CR dimension 1 and ofcodimension 3 has equations of such a form.Assuming V S (E 4 ) to be twin solvable and having covering submanifoldof solutions (see Definition 10.17), it may be verified (for M ⊂ C 3 , see[Be1997]) that at a Zariski-generic point, its equations are of the form:(8.3)y 1 = b 1 + xa + O(x 2 ) + O(b 1 ) + O(b 2 ),y 2 = b 2 + xa(x + a) + O(x 3 ) + O(b 1 ) + O(b 2 ).The model has zero remainders with associated system(8.4) y 2 1 = 2xy1 1 + (y1 1 )2 , y 1 2 = 0,the third equation y2 2 = 2 y1 1 being obtained by differentiating the first.We may put the submanifold in partial normal form. Proceeding asin [BES2005], some partial normalizations belonging to G v,p yield:(8.5)y 1 = b 1 + ax + a 2[ Π 1 3,2(b) x 3 + Π 1 4,2(b) x 4 + · · ·] + O(a 3 x 2 ),y 2 = b 2 + a [ x 2 + Π 2 4,1(b) x 4 + · · ·] + a 2[ x + Π 2 3,2(b) x 3 + · · ·] + O(a 3 x 2 ).Redifferentiating, we get an appropriate, partially normalized system (E 4 ):(8.6) ⎧⎪⎨y1 2 = y1( ) 1 2x + g1+ (y1) 1 2( 1 + g 2) + (y1) 1 3 s + (y1) 1 4 s + (y1) 1 5 s + (y1) 1 6 R,y2 1 ⎪⎩= (y1 1 )2 h + (y1 1 )3 R,y2 2 = )1( y1 2 + g1x + (y11 ) 2( gx 2 + (2x + g1 )h ) + (y1 1 )3 r + (y1 1 )4 r + (y1 1 )5 r + (y1 1 )6 R,where, precisely:• g 1 , g 2 and h are functions of (x, y 1 , y 2 ) satisfying g j = O(xx) +O(y 1 ) + O(y 2 ), j = 1, 2 and h = O(x) + O(y 1 ) + O(y 2 );267