  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.
