Bifurcations in Integrable Hamiltonian Systemsnear Corank-One Singularities

The structure of a real-analytic integrable Hamiltonian system with three degrees of freedom in neighborhoods of compact two-dimensional singular orbits of the corresponding Hamiltonian action (i.e., semilocal singularities of rank 2 and corank 1 of the energy-momentum map) 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. We present standard polynomial Hamiltonians, which, together with linear first integrals, give a $$C^\infty $$ -left-right classification of energy-momentum maps in neighborhoods of compact two-dimensional orbits. Bifurcations of the semilocal and semiglobal structure of the Liouville foliation near degenerate orbits are investigated, including those with twisting resonances of the form $$\ell /d$$ with any resonance order $$ d\in \mathbb N$$ . It is shown that these bifurcations 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 of reduced systems under the corresponding bifurcations are constructed.