Let $H$ be a real analytic Hamiltonian with an elliptic Diophantine fixed point and satisfying a Kolmogorov non-degeneracy condition. Then, it admits arbitrarily small real analytic perturbations (defined on a fixed polydisk) that are integrable on (small depending on the perturbation) neighborhoods of the origin. The proof is based on deformations of compatible complex and symplectic structures and on a KAM scheme where at each step nonlinear Cousin problems are solved.