Bifurcations of magnetic geodesic flows on toric surfaces of revolution

We study magnetic geodesic flows invariant under rotations on the 2-torus. The dynamical system is given by a generic pair of $2\pi$-periodic functions $(f,\Lambda)$, where the function $\Lambda$ takes values in a circle if the magnetic field is not exact. Topology of the Liouville fibration of the given integrable system near its singular orbits and singular fibers is decribed. Types of these singularities are computed. Topology of the Liouville fibration on regular 3-dimensional isoenergy manifolds is described by computing the Fomenko-Zieschang invariant. It is shown that Liouville fibrations for geodesic flow and non-exact magnetic geodesic flow on any isoenergy manifold have different topology. All possible bifurcation diagrams of the momentum maps of such integrable systems are described.