  Nowadays it is widely accepted that the evolution of the universe was driven
by some scalar degrees of freedom both on its early stage and at present. The
corresponding cosmological models often involve some scalar fields introduced
ad hoc. In this paper we cultivate a different approach, which is based on a
derivation of new scalar degrees of freedom from fundamental modifications of
Einstein's gravity. In elaboration of our previous work, we here investigate
properties of the dilaton-scalar gravity obtained by dimensional reductions of
a recently proposed affine generalized gravity theory. We show that these
models possess the same symmetries as related models of GR with ordinary scalar
fields. As a result, for a rather general class of dilaton-scalar gravity
models we construct additional first integrals and formulate an integral
equation well suited for solving by iterations.
