Classification of four-dimensional integrable Hamiltonian systems and Poisson actions of R(2) in extended neighbourhoods of simple singular points .3. Realization

The classification of integrable Hamiltonian systems in extended neighbourhoods of simple singular points, initiated in [1] and [2], is completed. This paper is a direct continuation of [1] and [2] and uses all the concepts, definitions, and results presented there. As promised in [2], the aim here is to realize all admissible types of the invariants describing the iso-energetic equivalence of integrable Hamiltonian systems with two degrees of freedom in extended neighbourhoods of simple singular points. In doing that, all admissible types of Poisson actions of the group R(2) on a four-dimensional symplectic manifold are also realized. Some of the examples constructed are of a purely illustrative nature, while others are more meaningful and occur in applications.