Chapter 9

Liouville Classication of Integrable

Systems with Two Degrees of Freedom

in Four-Dimensional Neighborhoods

of Singular Points

In this chapter, we present the results obtained by L. M. Lerman, Ya. L. Umanski

[212], [213], A. V. Bolsinov [44], V. S. Matveev [220], [221], Nguyen Tien Zung

[258], [261]. We shall follow the general idea of our book: try to present all facts from

the uniform viewpoint of the theory of topological invariants of integrable systems.

The preceding chapters were devoted to studying an integrable Hamiltonian system

on a three-dimensional isoenergy manifold. Here we wish to discuss its behavior

on a four-dimensional symplectic manifold. We shall mainly be interested in

the topological structure of the corresponding Liouville foliation.