The topology of Lagrangian foliations of integrable systems with hyperelliptic Hamiltonian

We study the integrable Hamiltonian systems
$$ (\mathbb C^2,\operatorname{Re}(dz\wedge dw),H=\operatorname{Re}f(z,w)) $$
with the additional first integral $F=\operatorname{Im}f$ which correspond to the complex Hamiltonian systems $(\mathbb C^2,dz\wedge dw,f(z,w))$ with a hyperelliptic Hamiltonian $f(z,w)=z^2+P_n(w)$, $n\in\mathbb N$. For $n\geqslant3$ the system has incomplete flows on any Lagrangian leaf $f^{-1}(a)$. The topology of the Lagrangian foliation of such systems in a small neighbourhood of any leaf $f^{-1}(a)$ is described in terms of the number $n$ and the combinatorial type of the leaf—the set of multiplicities of the critical points of the function $f$ that belong to the leaf. For odd $n$, a complex analogue of Liouville's theorem is obtained for those systems corresponding to polynomials $P_n(w)$ with simple real roots. In particular, a set of complex canonical variables analogous to action-angle variables is constructed in a small neighbourhood of the leaf $f^{-1}(0)$.
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