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