Bifurcations in Integrable Hamiltonian Systems near Corank-One Singularities

The structure of a real-analytic integrable Hamiltonian system with three degrees offreedom in neighborhoods of compact two-dimensional singular orbits of the correspondingHamiltonian action (i.e., semilocal singularities of rank 2 and corank 1 of the energy-momentummap) is studied. In integrable systems, such orbits usually occur in two-parameter families.Therefore, changes in the structure of the Liouville foliation are possible along the families. Wepresent standard polynomial Hamiltonians, which, together with linear first integrals, give a \(C^\infty\) -left-right classification of energy-momentum mapsin neighborhoods of compact two-dimensional orbits. Bifurcations of the semilocal and semiglobalstructure of the Liouville foliation near degenerate orbits are investigated, including those withtwisting resonances of the form \(\ell /d\) with anyresonance order \(d\in \mathbb N\) . It is shown that thesebifurcations are structurally stable with respect to analytic integrable perturbations of the system.In all cases, the bifurcation diagrams of energy-momentum mappings and the phase portraits ofreduced systems under the corresponding bifurcations are constructed.