Periodic solutions of Hamiltonian systems coupling twist with an isochronous center

We extend the Poincaré--Birkhoff Theorem to a Hamiltonian system which couples two systems with fairly different behaviors; the first one involves a twist assumption, while the second one is generated from a nonresonant isochronous center. By a suitable change of variables we modify the second system into a perturbation of a nonresonant linear one, and then prove that there exist multiple periodic solutions.