Formulation of the Results in Four Dimensions

We shall present mainly the new results obtained by the author recently. We begin with remarks of qualitative character. 

It is natural to start with the following question How is the fact of integrability of a Hamiltonian system associated with the topology of the phase (or configuration) manifold  

In the present chapter, we answer some of these questions. 

For the sake of simplicity, we begin with the four-dimensional case, although we have proved almost all the results formulated here in the multidimensional case as weU. This will be discussed in 2. 

In recent years many new results have been obtained concerning integration of Hamiltonian systems on symplectic manifolds. See, for instance, the survey in [143], [188]. Of particular interest in this connection is the discovery of stable closed integral trajectories of integrable systems. Such trajectories correspond to periodic motions of the system. 

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