The topology and isochronicity on complex Hamiltonian systems with homogeneous nonlinearities

In this paper, we study the Hamiltonian vector fields with homogeneous nonlinear parts on $C^2$. Firstly, we present a series of topological properties of the polynomial Hamiltonian function, with a particular focus on the characteristics of critical points and non-trivial cycles that vanish at infinity. Secondly, we use these topological properties to derive a complete set of necessary and sufficient conditions of isochronous centers for this class of systems of any degree. These conditions indicate that the isochronous center variety has two different components in the coefficient space of the nonlinear parts and each component is the intersection of several hyperplanes.