Dilaton-scalar models in the context of generalized affine gravity theories: Their properties and integrability

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.