Incomplete integrable Hamiltonian systems with complex polynomial Hamiltonian of small degree

Complex Hamiltonian systems with one degree of freedom on C2 with the standard symplectic structure ωC=dz and dw and a polynomial Hamiltonian function f=z2+Pn(w), n=1,2,3,4, are studied. Two Hamiltonian systems (Mi, ReωC,i, Hi=Refi), i=1,2, are said to be Hamiltonian equivalent if there exists a complex symplectomorphism M1→M2 taking the vector field sgradH1 to sgradH2. 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. Bibliography: 9 titles.