Incomplete integrable Hamiltonian systems with complex polynomial Hamiltonian of small degree

Complex Hamiltonian systems with one degree of freedom on C(2) with the standard symplectic structure omega(C) = dz boolean AND dw and a polynomial Hamiltonian function f = z(2) + P(n)(w), n = 1, 2, 3, 4, are studied. Two Hamiltonian systems (M(i), Re omega(C,i), H(i) - Re f(i)), i - 1, 2, are said to be Hamiltonian equivalent if there exists a complex symplectomorphism M(1) -> M(2) taking the vector field sgrad H(1) to sgrad H(2). Hamiltonian equivalence classes of systems are described in the case n = 1, 2, 3, 4, a completed system is defined for n = 3, 4, and it is proved that it is Liouville integrable as a real Hamiltonian system. By restricting the real action-angle coordinates defined for the completed system in a neighbourhood of any nonsingular leaf, real canonical coordinates are obtained for the original system.