Topological classification of non-compact 3-atoms with a circle action

For integrable Hamiltonian systems with two degrees of freedom we investigate the topology of the Liouville foliation in a 3-dimensional non-compact invariant neighborhood of a singular leaf. All the singularities of the system are supposed to be non-degenerate. In the case when all the leaves of the Liouville foliation are compact, this problem is already solved: the well-known A. T. Fomenko theorem states that any non-degenerate 3-dimensional singularity (3-atom) is an $S^1$-fibration of the special type (Seifert fibration) over a 2-dimensional singularity (2-atom). Thus, the problem of the topological classification of 3-atoms is reduced to the significantly more simple classification problem for 2-atoms (i. e. singularities of foliations determined by Morse functions on 2-surfaces). The latter problem is well-studied in the framework of the Fomenko classification theory for integrable systems. In the non-compact case, the set of all 3-atoms becomes much richer. That is why we consider only 3-atoms satisfying the following conditions: completeness of the Hamiltonian flows generated by the first integrals of the system, finiteness of the number of orbits of the Hamiltonian $\mathbb{R}^2$-action on the singular leaf, and existence among these orbits of a non-contractible one. Under these restrictions, we proof that the 3-atom admits a Hamiltonian locally free $S^1$-action preserving the leaves of the Liouville foliation. As a corollary, we obtain the analogue of the Fomenko theorem and thus reduce the classification problem for non-compact 3-atoms satisfying the above conditions to the similar classification problem for non-compact 2-atoms that we solved earlier.