Hamiltonian Systems with Poisson Commuting Integrals

abelian algebra abelian subalgebra action-and-angle variables assume that h Borels lemma c(hº Cartan subalgebra centermanifolds complex standard base const construct coordinates denote dimension 2n elliptic C.S.A. exist smooth functions exists a smooth flat foliation follows formal series functions V1 h is defined Hamiltonian system Hence implicit function theorem implies induction integrable systems invariant Let h Let q levelsets Lie algebra local diffeomorphism matrix mixed pair modulo Moreover Moser non-degenerate non-singular Poisson commuting functions POISSON COMMUTING INTEGRALS proof goes Proposition H quadratic differential quadratic functions real analytic Remark singular manifolds singular space smooth functions defined solution solve the equations Sp(V span stationary point submanifold symplectic base symplectic chart symplectic diffeomorphism symplectic manifold symplectic mapping symplectic structure symplectic submanifold symplectic subspaces symplectic vector space vanishes quadratically vector verify