  The volumes, spectra and geodesics of a recently constructed infinite family
of five-dimensional inhomogeneous Einstein metrics on the two $S^3$ bundles
over $S^2$ are examined. The metrics are in general of cohomogeneity one but
they contain the infinite family of homogeneous metrics $T^{p,1}$. The geodesic
flow is shown to be completely integrable, in fact both the Hamilton-Jacobi and
the Laplace equation separate. As an application of these results, we compute
the zeta function of the Laplace operator on $T^{p,1}$ for large $p$. We
discuss the spectrum of the Lichnerowicz operator on symmetric transverse
tracefree second rank tensor fields, with application to the stability of
Freund-Rubin compactifications and generalised black holes.
