  For any toric automorphism with only real eigenvalues a Riemannian metric
with an integrable geodesic flow on the suspension of this automorphism is
constructed. A qualitative analysis of such a flow on a three-solvmanifold
constructed by the authors in math.DG/9905078 is done. This flow is an example
of the geodesic flow, which has vanishing Liouville entropy and, moreover, is
integrable but has positive topological entropy. The authors also discuss some
open problems on integrability of geodesic flows and related subjects.
