Integrable Perturbations of Polynomial Hamiltonian Systems

We consider a Hamiltonian system on the symplectic space $$( \mathbb{R} ^{2n}, dy\wedge dx)$$ with a real-analytic Hamiltonian $$H : \mathbb{R} ^{2n}\to \mathbb{R} $$ . 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 $$F: \mathbb{R} ^{2n}\to \mathbb{R} $$ such that (1) $$F = O\big( (|x|+|y|)^{M+1} \big)$$ at the origin, (2) the system with Hamiltonian $$H+F$$ is completely integrable in $$ \mathbb{R} ^{2n}$$ .