ИСТИНА

We study singular Lagrangian fibrations that arise in typical integrable Hamiltonian systems with 3 degrees of freedom. We study semi-local singularities of such Lagrangian fibrations, i.e., the topological structure of the singular Lagrangian fibration near compact singular orbits of the Hamiltonian R^3-action generated by the energy-momentum map. We describe 1-dimensional compact singular orbits with hyperbolic m:n-resonance (0<m\le n), that are similar to the Hamiltonian Hopf bifurcation with elliptic m:n-resonance (0<m\le|n|) stydied by Duistermaat (1984) and van der Meer (1985). We prove that these singularities are structurally stable under real-analytic perturbations of the energy-momentum map. Duistermaat (1984) studied a Hamiltonian Hopf bifurcation of a generic Hamiltonian system with 2 degrees of freedom and 1 parameter, near an equilibrium with elliptic m:n-resonance. He proved that periodic orbits near such an equilibrium are diffeomorphic images of periodic orbits of the corresponding integrable Hamiltonian system. We extend this rusult to the case of hyperbolic m:n-resonance.