Integrable Hamiltonian system with two degrees of freedom. The topological structure of saturated neighbourhoods of points of focus-focus and saddle-saddle type

One of the key problem in Hamiltonian mechanics is to describe the behaviour of integrable Hamiltonian system with two degrees of freedom in neighbourhoods of singular leaves of the Liouville foliation. In 1988 Lerman and Umanskii announced a classification theorem for such systems in the case when the singular leaf contains one point of rank zero. The proof was published in 1992 and 1993. In the solution of classical problems in physics and mechanics, integrable Hamiltonian systems arise which have a singular leaf with several points of rank zero. This paper concerns the topological classification of integrable Hamiltonian systems in the neighbourhood of singular leaves that contain an arbitrary number of points of rank zero.