  The paper presents a generalization and further development of our recent
publications where solutions of the Klein-Fock-Gordon equation defined on a few
particular $D=(2+1)$-dim static space-time manifolds were considered. The
latter involve toy models of 2-dim spaces with axial symmetry, including
dimension reduction to the 1-dim space as a singular limiting case.
  Here the non-static models of space geometry with axial symmetry are under
consideration. To make these models closer to physical reality, we define a set
of "admissible" shape functions $\rho(t,z)$ as the $(2+1)$-dim Einstein
equations solutions in the vacuum space-time, in the presence of the
$\Lambda$-term, and for the space-time filled with the standard "dust". It is
curious that in the last case the Einstein equations reduce to the well-known
Monge-Amp\`{e}re equation, thus enabling one to obtain the general solution of
the Cauchy problem, as well as a set of other specific solutions involving one
arbitrary function. A few explicit solutions of the Klein-Fock-Gordon equation
in this set are given.
  An interesting qualitative feature of these solutions relates to the
dimension reduction points, their classification, and time behavior. In
particular, these new entities could provide us with novel insight into the
nature of P- and T-violation, and of Big Bang. A short comparison with other
attempts to utilize dimensional reduction of the space-time is given.
