Classification of four-dimensional integrable Hamiltonian systems and Poisson actions of $\mathbb R^2$ in extended neighbourhoods of simple singular points. III. 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 $\mathbb R^2$ 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.