The center conditions for nearly hamiltonian systems

We consider the center-focus problem for ODE systems close to the plane Hamiltonian ones. We suggest a method to obtain the necessary conditions for the center, which at the same time gives the asymptotics of the Poincare mapping in case of the focus. A perturbed Hamiltonian system having the Newton open polygon with two edges is considered for the first time. It is shown that such cases are reduced to ODE systems on Riemann surfaces.