Isochronicity of plane polynomial Hamiltonian systems

We study isochronous centres of plane polynomial Hamiltonian systems, and more generally, isochronous Morse critical points of complex polynomial Hamiltonian functions. Our first result is that if the Hamiltonian function H is a non-degenerate semi-weighted homogeneous polynomial, then it cannot have an isochronous Morse critical point, unless the associate Hamiltonian system is linear, that is to say H is of degree two. Our second result gives a topological obstruction for isochronicity. Namely, let gamma(h) be a continuous family of one-cycles contained in the complex level set H-1(h), and vanishing at an isochronous Morse critical point of H, as h --> 0. We prove that if H is a good polynomial with only simple isolated critical points and the level set H-1(0) contains a single critical point, then gamma(h) represents a zero homology cycle on the Riemann surface of the algebraic curve H-1(h). We give several examples of 'non-trivial' complex Hamiltonians with isochronous Morse critical points and explain how their study is related to the famous Jacobian conjecture.