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

TL;DR

This paper extends Liouville's theorem to certain integrable Hamiltonian systems with incomplete flows, defining a new type of Liouville fibration using flat polygons and analyzing systems defined by complex polynomial vector fields.

Contribution

It introduces an analogue of Liouville's theorem for systems with incomplete flows and describes the geometric structure of their Liouville fibrations.

Findings

Defined a canonical Liouville fibration using flat polygons.

Analyzed geometric properties of Liouville fibrations for polynomial Hamiltonian systems.

Extended classical integrability results to systems with incomplete flows.

Abstract

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 $v=(-\partial f/\partial w, \partial f/\partial z)$ on ${\mathbb C}^2$ where $f=f(z,w)$ is a complex polynomial in 2 variables, geometric properties of Liouville fibrations are described.