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

291are obtained by equating to zero all (m+p M )×(m+p M ) minors of Jac Q ∞and all (m + n M ) × (m + n M ) minors of Jac Q∞ ∗ , such that for everypoint p = (x p , y p , a p , b p ) ∉ Σ M , there exists a local change of coordinatesrespecting the separation of the variables (x, y) and (a, b)(9.9) (x, y, a, b) ↦→ ( ϕ(x, y), h(a, b) ) =: (x ′ , y ′ , a ′ , b ′ )by which M is transformed to a submanifold M ′ centered and localized atp ′ = p having equations(9.10) y ′ = Π ′ (x ′ , a ′ , b ′ ) and dually b ′ = Π ′∗ (a ′ , x ′ , y ′ )with Π ′ and Π ′∗ independent of((9.11) x′nM +1 , . . ., n)x′and of(a′pM +1 , . . ) .,a′ p .So M ′ , may be considered to be living in K n Mx ′ × K m y ′ × Kp Ma ′ × K m b ′ andin such a smaller space, at p ′ = p, it is solvable both with respect to theparameters and to the variables.Interpretation: by forgetting some innocuous variables, at a Zariskigenericpoint, any M is both solvable with respect to the parameters andto the variables. These two assumptions will be held up to the end of thisPart I.9.12. Dual system (E ∗ ) and isomorphisms SYM(E ) ≃SYM ( V S (E ) ) = SYM ( V S (E ∗ ) ) ≃ SYM(E ∗ ). To a system(E ), we associate its submanifold of solutions M := V S (E ). Assumingit to be solvable with respect to the variables and proceeding as in §2.10,we can derive a dual system of completely integrable partial differentialequations of the form((E ∗ ) b j a γ(a) = Gj γ a, b(a), ( b j(l) (a) ) ),a δ(l) 1lnwhere (j, γ) ≠ (j, 0) and ≠ (j(l), δ(l)). Its submanifold of solutionsV S (E ∗ ) ≡ V S (E ) has equations dual to those of V S (E ).Theorem 9.13. Under the assumption of twin solvability, we have:(9.14) SYM(E ) ≃ SYM ( V S (E ) ) = SYM ( V S (E ∗ ) ) ≃ SYM(E ∗ ),through L ←→ L + L ∗ = L ∗ + L ←→ L ∗ .§10. FUNDAMENTAL PAIR OF FOLIATIONS AND COVERING PROPERTY10.1. Fundamental pair of foliations on M . As in §2, let (E ) and M =V S (E ) be defined by y = Π(x, a, b) or dually by b = Π ∗ (a, x, y). Abbreviate(10.2) z := (x, y) and c := (a, b).