The Conley–Zehnder index and the saddle-center equilibrium

We consider Hamiltonian systems with two degrees of freedom. We suppose the existence of a saddle-center equilibrium in a strictly convex component S of its energy level. Moser's normal form for such equilibriums and a theorem of Hofer, Wysocki and Zehnder are used to establish the existence of a periodic orbit in S with several topological properties. We also prove the explosion of the Conley–Zehnder index of any periodic orbit that passes close to the saddle-center equilibrium.