  We classify Einstein metrics on $\mathbb{R}^4$ invariant under a
four-dimensional group of isometries including a principal action of the
Heisenberg group. The metrics are either Ricci-flat or of negative Ricci
curvature. We show that all of the Ricci-flat metrics, including the simplest
ones which are hyper-K\"ahler, are incomplete. By contrast, those of negative
Ricci curvature contain precisely two complete examples: the complex hyperbolic
metric and a metric of cohomogeneity one known as the one-loop deformed
universal hypermultiplet.
