  We prove that on closed Riemannian manifolds with infinite abelian, but not
cyclic, fundamental group, any isometry that is homotopic to the identity
possesses infinitely many invariant geodesics. We conjecture that the result
remains true if the fundamental group is infinite cyclic. We also formulate a
generalization of the isometry-invariant geodesics problem, and a
generalization of the celebrated Weinstein conjecture: on a closed contact
manifold with a selected contact form, any strict contactomorphism that is
contact-isotopic to the identity possesses an invariant Reeb orbit.
