  We investigate the possibility of having an event horizon within several
classes of metrics that asymptote to the maximally supersymmetric IIB plane
wave. We show that the presence of a null Killing vector (not necessarily
covariantly constant) implies an effective separation of the Einstein equations
into a standard and a wave component. This feature may be used to generate new
supergravity solutions asymptotic to the maximally supersymmetric IIB plane
wave, starting from standard seed solutions such as branes or intersecting
branes in flat space. We find that in many cases it is possible to preserve the
extremal horizon of the seed solution. On the other hand, non-extremal
deformations of the plane wave solution result in naked singularities. More
generally, we prove a no-go theorem against the existence of horizons for
backgrounds with a null Killing vector and which contain at most null matter
fields. Further attempts at turning on a nonzero Hawking temperature by
introducing additional matter have proven unsuccessful. This suggests that one
must remove the null Killing vector in order to obtain a horizon. We provide a
perturbative argument indicating that this is in fact possible.
