  We present a systematic attempt at classification of supersymmetric M-theory
vacua with zero flux; that is, eleven-dimensional lorentzian manifolds with
vanishing Ricci curvature and admitting covariantly constant spinors. We show
that there are two distinct classes of solutions: static spacetimes
generalising the Kaluza-Klein monopole, and non-static spacetimes generalising
the supersymmetric wave. The classification can be further refined by the
holonomy group of the spacetime. The static solutions are organised according
to the holonomy group of the spacelike hypersurface, whereas the non-static
solutions are similarly organised by the (lorentzian) holonomy group of the
spacetime. These are subgroups of the Lorentz group which act reducibly yet
indecomposably on Minkowski spacetime. We present novel constructions of
non-static vacua consisting of warped products of d-dimensional pp-waves with
(11-d)-dimensional manifolds admitting covariantly constant spinors. Our
construction yields local metrics with a variety of exotic lorentzian holonomy
groups. In the process, we write down the most general local metric in d<6
dimensions describing a pp-wave admitting a covariantly constant spinor.
Finally, we also discuss a particular class of supersymmetric vacua with
nonzero four-form obtained from the previous ones without modifying the
holonomy of the metric. This is possible because in a lorentzian spacetime a
metric which admits parallel spinors is not necessarily Ricci-flat, hence
supersymmetric backgrounds need not satisfy the equations of motion.
