  We introduce generalized dimensional reductions of an integrable
1+1-dimensional dilaton gravity coupled to matter down to one-dimensional
static states (black holes in particular), cosmological models and waves. An
unusual feature of these reductions is the fact that the wave solutions depend
on two variables - space and time. They are obtained here both by reducing the
moduli space (available due to complete integrability) and by a generalized
separation of variables (applicable also to non integrable models and to higher
dimensional theories). Among these new wave-like solutions we have found a
class of solutions for which the matter fields are finite everywhere in
space-time, including infinity.
  These considerations clearly demonstrate that a deep connection exists
between static states, cosmologies and waves. We argue that it should exist in
realistic higher-dimensional theories as well. Among other things we also
briefly outline the relations existing betweenthe low-dimensional models that
we have discussed hereand the realistic higher-dimensional ones.
  This paper develops further some ideas already present in our previous
papers. We briefly reproduce here (without proof) their main results in a more
concise form and give an important generalization.
