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 non-degenerate 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) $F=O((\mathrm{\mid }x\mathrm{\mid }+\mathrm{\mid }y\mathrm{\mid }{)}^{M+1})$ at the origin, (2) the system with Hamiltonian $H+F$ is completely integrable in ${\mathbb{R}}^{2n}$.