The complexity of integrable Hamiltonian systems on a prescribed three-dimensional constant-energy submanifold

This paper is devoted to a description of $Q$-regions, i.e., domains in the molecular table of Fomenko that are filled with integrable systems with constant energy surfaces $Q$ that occur most frequently in physics. Namely, the $Q$-regions for $Q=S^3$, $\mathbf RP^3$, $S^1\otimes S^2$, $T^3$, and $\overset l\#S^1\otimes S^2$ are computed explicitly. The $Q$-regions for an arbitrary three-dimensional constant energy submanifold $Q$ are determined up to a finite number of points. These results make it possible to predict the topological properties of integrable Hamiltonian systems as yet not discovered in physics. The concepts of the order of torsion of integrable Hamiltonian systems and of a minimal system are also introduced, and the connection between these concepts and the concepts of complexity of systems and complexity of three-manifolds due to Matveev is indicated.