  Supersymmetric domain-wall spacetimes that lift to Ricci-flat solutions of
M-theory admit generalized Heisenberg (2-step nilpotent) isometry groups. These
metrics may be obtained from known cohomogeneity one metrics of special
holonomy by taking a "Heisenberg limit", based on an In\"on\"u-Wigner
contraction of the isometry group. Associated with each such metric is an
Einstein metric with negative cosmological constant on a solvable group
manifold. We discuss the relevance of our metrics to the resolution of
singularities in domain-wall spacetimes and some applications to holography.
The extremely simple forms of the explicit metrics suggest that they will be
useful for many other applications. We also give new but incomplete
inhomogeneous metrics of holonomy SU(3), $G_2$ and Spin(7), which are $T_1$,
$T_2$ and $T_3$ bundles respectively over hyper-K\"ahler four-manifolds.
