Isochronicity and linearizability of planar polynomial Hamiltonian systems

In this paper we study isochronicity and linearizability of planar polynomial Hamiltonian systems. First we prove a theorem which supports a negative answer to the following open question stated by Jarque and Villadelprat in [15]: Do there exist planar polynomial Hamiltonian systems of even degree having an isochronous center? Additionally we obtain some conditions for linearizability of complex cubic Hamiltonian systems.