Integrability of geodesic flows and isospectrality of Riemannian manifolds

We construct a pair of compact, eight-dimensional, two-step Riemannian nilmanifolds $M$ and $M'$ which are isospectral for the Laplace operator on functions and such that $M$ has completely integrable geodesic flow in the sense of Liouville, while $M'$ has not. Moreover, for both manifolds we analyze the structure of the submanifolds of the unit tangent bundle given by to maximal continuous families of closed geodesics with generic velocity fields. The structure of these submanifolds turns out to reflect the above (non)integrability properties. On the other hand, their dimension is larger than that of the Lagrangian tori in $M$, indicating a degeneracy which might explain the fact that the wave invariants do not distinguish an integrable from a nonintegrable system here. Finally, we show that for $M$, the invariant eight-dimensional tori which are foliated by closed geodesics are dense in the unit tangent bundle, and that both $M$ and $M'$ satisfy the so-called Clean Intersection Hypothesis.