Coupling linearity and twist: an extension of the Poincaré–Birkhoff theorem for Hamiltonian systems

We provide an extension of the Poincaré-Birkhoff Theorem for systems coupling linear components with twisting components. Applications are given both to weakly coupled Hamiltonian systems where, e.g., a superlinear or sublinear behaviour is assumed in the nonlinear part of the coupling in order to recover the needed twist conditions, and to local perturbations of superintegrable systems, showing the survival of a number of periodic solutions from a lower-dimensional torus.