Construction of integrable cosmological models with non-minimally coupled scalar fields

We consider the relation between integrable cosmological models having minimally and non-minimally coupled scalar fields. We show how to obtain general solutions for some flat Friedmann universes filled with a scalar field in induced gravity models and models including the Hilbert-Einstein curvature term plus a scalar field conformally coupled to gravity. The explicit forms of the self-interaction potentials for seven exactly solvable models are presented here.