Non-closed isometry-invariant geodesics

Let c be a non-closed and bounded geodesic in a complete Riemannian manifold M. Assume that c is invariant under an isometry A of M and that c is not contained in the set of fixed points of A. We prove that, for some , the geodesic flow line A < corresponding to c is dense in a k-dimensional torus N embedded in TM. In particular, every geodesic with initial vector in N is A-invariant.