  We consider a cosmological model with two scalar fields minimally coupled to
gravity which have a mixed kinetic term. Hence, Chiral cosmology is included in
our analysis. The coupling function and the potential function, which depend on
one of the fields, characterize the model we study. We prove the existence of
exact solutions that are of special interest for the cosmological evolution.
Furthermore, we provide with a methodology that relates the scale factor
behaviour to the free functions characterizing the scalar field kinetic term
coupling and potential. We derive the necessary conditions that connect these
two functions so that the relative cosmological solutions can be admitted. We
find that unified dark matter and dark energy solutions are allowed by the
theory in various scenarios involving the aforementioned functions.
