Nondegenerate systems and generic properties of the integrable Hamiltonian systems (Q1303355)

In the paper the author clarifies whether the nondegenerate systems are generic among all integrable systems. This question is important because the nondegenerate systems can be described up to topological equivalence by the Fomenko-Zieschang isoenergetic invariant. It has been proved that every integrable nondegenerate Hamiltonian system can be made integrable degenerate on the given energetic level by a small perturbation in the \textit{weak} metric, i.e. degenerate systems are dense in this topology. Besides under the assumptions that a Hamiltonian system has a nondegenerate integral \(f\) on \(Q_{h}=\{H=h\}\), all critical manifolds of \(f|_{Q_{h}}\) are circles and there is only one circle on every critical level of \(f\), the author proves that the Fomenko-Zieschang isoenergetic invariant of the system on \(Q_{h}\) does not change under small perturbations in the \textit{strong} metric.