| Contents |
| In a non-reducible integrable Hamiltonian system with three degrees of freedom, we can present the set of the critical points of the momentum mapping as a union of the phase spaces of Hamiltonian systems with less number of degrees of freedom. In such critical subsystems, manifolds of co-dimension 1 can exist, on which the induced symplectic structure degenerates. Sometimes the topology of a critical subsystem does not notice such degeneration. In this case the theory of the Fomenko-Zieschang topological invariants can be applied. The corresponding example is the integrable system with two degrees of freedom found by O.I.Bogoyavlensky in the dynamics of a heavy magnet. In other problems we come across bifurcations which are impossible in the systems without singularities of the symplectic form. Moreover, we present an example of a critical subsystem with two degrees of freedom having non-orientable phase space. In this system, new types of loop molecules, non-orientable 3-atoms and bifurcations arise. We classify 3-dimensional isoenergetic manifolds which are $S^1$-bundles over orientable or non-orientable 2-dimensional surfaces. The investigation is carried out with the help of algebraic separation of variables found for this system. Such separating gives rise to a universal algorithm of calculating exact topological invariants of the system in the form of gluing matrices. |
|