Integrable systems with incomplete flows and Newton's polygons

Consider an integrable Hamiltonian system on a 4-dimensional symplectic manifold, given by a pair of smooth functions in involution. Due to the Liouville theorem, compact connected regular common level sets of such functions (called Liouville leaves) are homeomorphic to the $2$-torus. As is well-known, non-compact Liouville leaves can be homeomorphic to a sphere with any number of handles or punctures. In 1988, H. Flaschka suggested a wide class of such systems given by vector fields $v=(-df/dw,df/dz)$ on $\mathbb C^2$ with the symplectic $2$-form $\operatorname{Re}(dz\wedge dw)$, where $f=f(z,w)$ is a non-constant complex polynomial in $2$ variables. Recently, in a joint work with T. A. Lepskii, we obtained an analogue of the Liouville theorem for systems of the Flaschka class corresponding to hyperelliptic polynomials $f(z,w)=z^2+P(w)$. In this talk, we will suggest an analogue of the Liouville theorem for a wider class of integrable Hamiltonian systems with incomplete flows, which includes the class considered by Flaschka, and give a combinatorial-topological description of a family of “standard” Liouville foliations.