Scalar-tensor Cosmologies can be dealt under the standard of the Hojman
conservation theorem that allows to fix the form of the coupling F(ϕ), of
the potential V(ϕ) and to find out exact solutions for related
cosmological models. Specifically, the existence of a symmetry transformation
vector for the equations of motion gives rise to a Hojman conserved quantity on
the corresponding minisuperpace and exact solutions for the cosmic scale factor
a and the scalar field ϕ can be achieved. In particular, we take
advantage of the fact that minimally coupled solutions, previously obtained in
the Einstein frame, can be conformally transformed in non-minimally coupled
solutions in the Jordan frame. Some physically relevant examples are worked
out.Comment: 6 pages, 4 figures, to appear in Phys. Lett.