  We consider the dilaton gravity models derived by reductions of generalized
theories of gravity and study one-dimensional dynamical systems simultaneously
describing cosmological and static states in any gauge. Our approach is fully
applicable to studying static and cosmological solutions in multidimensional
theories and also in general one-dimensional dilaton - scalaron gravity models.
We here focus on general and global properties of the models, on seeking
integrals, and on analyzing the structure of the solution space. We propose
some new ideas in this direction and derive new classes of integrals and new
integrable models.
