Rigidity for periodic magnetic fields

We study the motion of a charge on a conformally flat Riemannian torus in the presence of a magnetic field. We prove that for any non-zero magnetic field there always exist orbits of this motion which have conjugate points. We conjecture that the restriction of conformal flatness of the metric is not essential for this result. This would provide a 'twisted' version of the recent generalization of Hopf's rigidity result obtained by Burago and Ivanov.