Integrable Perturbations of Polynomial Hamiltonian Systems

We consider a Hamiltonian system on the symplectic space ( R ²ⁿ, dy dx) with a real-analytic Hamiltonian H : R ²ⁿ→ R . We assume that the system has a nondegenerate equilibrium position at the origin. Under some nonresonance assumptions, we prove the following. For any positive integer M , there exists a real-analytic function F: R ²ⁿ→ R such that (1) F = O( (|x|+|y|)M+1 ) at the origin, (2) the system with Hamiltonian $$H+F$$ is completely integrable in R ²ⁿ .