Bifurcations of integrable mechanical systems with magnetic field on surfaces of revolution

On a surface homeomorphic to $2$-sphere, we study a natural mechanical system with a magnetic field that is invariant under the $S^1$-action. For singular points of rank $0$ of the momentum mapping, a criterion for non-degeneracy is obtained, the type of non-degenerate singular points (center-center and focus-focus) is determined, bifurcations of typical degenerate singular points are described (integrable Hamiltonian Hopf bifurcation of two types). For families of singular circles of rank $1$ of the momentum mapping (consisting of relative equilibriums of the system) their parametric representation is obtained, nondegeneracy criterion is proved, the type of nondegenerate (elliptic and hyperbolic) and typical degenerate (parabolic) singular circles is determined. The parametric representation of the bifurcation diagram of the momentum mapping is obtained. Geometric properties of the bifurcation diagram and the bifurcation complex are described in the case when the functions defining the system are in general position. The topology of nonsingular isoenergy $3$-dimensional manifolds is determined, the topology of the Liouville foliation on them is described up to the rough Liouville equivalence (in terms of Fomenko's atoms and molecules). The “splitting” hyperbolic singularities of rank $1$ are described, which are topologically unstable bifurcations of the Liouville foliation.