Real description of classical Hamiltonian dynamics generated by a complex potential

We show that the complex dynamics defined by an analytic potential v:C→C admits a description in terms of the phase space R4R4 equipped with an unconventional symplectic structure J. We obtain the general form of J   and devise an equivalent real description of the system that is based on the conventional symplectic structure on R4R4. In this description the real part of the complex Hamiltonian generates the dynamics and its complex part is an independent integral of motion rendering the system integrable.