The Liouville-type theorem for integrable Hamiltonian systems with incomplete flows

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For integrable Hamiltonian systems with two degrees of freedom whose Hamiltonian vector fields have incomplete flows, an analogue of the Liouville theorem is established. A canonical Liouville fibration is defined by means of an "exact" 2-parameter family of flat polygons equipped with certain pairing of sides. For the integrable Hamiltonian systems given by the vector field v=(-∂ f/∂ w, ∂ f/∂ z) on C² where f=f(z,w) is a complex polynomial in 2 variables, geometric properties of Liouville fibrations are described.