  Let $Q$ be a closed manifold admitting a locally-free action of a compact Lie
group $G$. In this paper we study the properties of geodesic flows on $Q$ given
by Riemannian metrics which are invariant by such an action. In particular, we
will be interested in the existence of geodesics which are closed up to the
action of some element in the group $G$, since they project to closed magnetic
geodesics on the quotient orbifold $Q/G$.
