We investigate cohomogeneity-one metrics whose principal orbit is an
Aloff-Wallach space SU(3)/U(1). In particular, we are interested in metrics
whose holonomy is contained in Spin(7). Complete metrics of this kind which are
not product metrics have exactly one singular orbit. We prove classification
results for metrics on tubular neighborhoods of various singular orbits. Since
the equation for the holonomy reduction has only few explicit solutions, we
make use of power series techniques. In order to prove the convergence and the
smoothness near the singular orbit, we apply methods developed by Eschenburg
and Wang. As a by-product of these methods, we find many new examples of
Einstein metrics of cohomogeneity one.Comment: 46 page
