Integrable Hamiltonian systems with positive topological entropy

In this paper we provide a class of integrable Hamiltonian systems on a three-dimensional Riemannian manifold whose flows have a positive topological entropy on almost all compact energy surfaces. As our knowledge, these are the first examples of C∞ Liouvillian integrable Hamiltonian flows with potential energy on a Riemannian manifold which has a positive topological entropy.