Incomplete integrable Hamiltonian systems with complex polynomial Hamiltonian of small degree

Complex Hamiltonian systems with one degree of freedom on $\mathbb C^2$ with the standard symplectic structure $\omega_\mathbb C=dz\wedge dw$ and a polynomial Hamiltonian function $f=z^2+P_n(w)$, $n=1,2,3,4$, are studied. Two Hamiltonian systems $(M_i,\,\operatorname{Re}\omega_{\mathbb C,i},\,H_i=\operatorname{Re}f_i)$, $i=1,2$, are said to be Hamiltonian equivalent if there exists a complex symplectomorphism $M_1\to M_2$ taking the vector field $\operatorname{sgrad}H_1$ to $\operatorname{sgrad}H_2$. 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.
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