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Topology of surfaces of constant energy of integrable Hamiltonian systems and barriers to integrability. (Russian) Zbl 0619.58023
Let \(M^ 4\) be a smooth four-dimensional symplectic manifold with Hamiltonian system \(v=S \text{grad} H\) (H.S.). Then the smooth integral f is called the Botts integral if the critical set of f is a smooth submanifold of M. The author obtains the complete topological classification of submanifolds of constant energy H on a manifold \(M^ 4\) with Botts integral f, and gives a canonical representation of such submanifolds as a connected sum of 3-manifolds of simple types (theorem 3). With the help of this result the author gets the lower estimate for the number of stable closed orbits of H.S. depending on \(H_ 1(M,{\mathbb{Z}})\) and some new topological barriers to complete integrability of H.S. with the help of Botts integrals.
Reviewer: V.B.Marenich

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems