On a surface homeomorphic to 2-sphere, we study a natural mechanical system with amagnetic field that is invariant under the $S^1$-action. For singular points of rank 0 of themomentum mapping, a criterion for non-degeneracy is obtained, the type of non-degeneratesingular points (center-center and focus-focus) is determined, bifurcations of typical degeneratesingular points are described (integrable Hamiltonian Hopf bifurcation of two types). For familiesof singular circles of rank 1 of the momentum mapping (consisting of relative equilibriums of thesystem) their parametric representation is obtained, nondegeneracy criterion is proved, the typeof nondegenerate (elliptic and hyperbolic) and typical degenerate (parabolic) singular circlesis determined. The parametric representation of the bifurcation diagram of the momentummapping is obtained. Geometric properties of the bifurcation diagram and the bifurcationcomplex 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 ofthe Liouville foliation on them is described up to the rough Liouville equivalence (in terms ofFomenko’s atoms and molecules). The “splitting” hyperbolic singularities of rank 1 are described,which are topologically unstable bifurcations of the Liouville foliation.