  After a brief exposition of the simplest class of affine theories of gravity
in multidimensional space-times with symmetric connections, we consider the
spherical and cylindrical reductions of these theories to two-dimensional
dilaton-vecton gravity (DVG) field theories. The distinctive feature of these
theories is the presence of a massive/tachyonic vector field (vecton) with
essentially nonlinear coupling to the dilaton gravity. In the massless limit,
the classical DVG theory can be exactly solved for a rather general coupling
depending only on the field tensor and the dilaton. We show that the vecton
field can be consistently replaced by a new effectively massive scalar field
(scalaron) with an unusual coupling to dilaton gravity (DSG). Then we
concentrate on considering the DVG models derived by reductions of D=3 and D=4
affine theories. In particular, we introduce the most general cylindrical
reductions that are often ignored. The main subject of our study is the static
solutions with horizons. We formulate the general conditions for the existence
of the regular horizons and find the solutions of the static DVG/DSG near the
horizons in the form of locally convergent power - series expansion. For an
arbitrary regular horizon, we find a local generalization of the Szekeres -
Kruskal coordinates. Finally, we consider one-dimensional integrable and
nonintegrable DSG theories with one scalar. We analyze simplest models having
three or two integrals of motion, respectively, and introduce the idea of a
`topological portrait' giving a unified qualitative description of static and
cosmological solutions of some simple DSG models.
