The topology of integral submanifolds of completely integrable Hamiltonian systems

It is proved that the class $(X)$ of three-dimensional closed compact manifolds that are constant energy surfaces of integrable (by means of a Bott integral) Hamiltonian systems coincides precisely with the class $(Q)$ of three-dimensional orientable manifolds admitting decomposition into “circular handles”. Fomenko previously proved the inclusion $(X)\subset(Q)$. An explicit geometric description is also given for modifications of Liouville tori in neighborhoods of nonorientable critical submanifolds of the moment mapping of an integrable system.
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Bibliography: 20 titles.