Typical integrable Hamiltonian systems on a four-dimensional symplectic manifold

We study the topology of integrable Hamiltonian systems with two degrees of freedom in the neighbourhood of a degenerate circle. Among all degenerate circles, the class of so-called generic degenerate circles is singled out. These circles cannot be removed from the symplectic manifold by a small perturbation of the Poisson action, and the system remains topologically equivalent to the unperturbed system in their neighbourhood. Moreover, if the unperturbed system has only Bott circles and generic degenerate circles, then, under the condition of simplicity, the perturbed system is globally topologically equivalent to it. It is proved that if an additional condition holds, then there is a small perturbation for which all degenerate circles are generic.