  Adopting Noether point symmetries, we classify and integrate dynamical
systems coming from Horndeski cosmologies. The method is particularly effective
both to select the form of Horndeski models and to derive exact cosmological
solutions. Starting from the Lagrangians selected by the Noether symmetries, it
is possible to derive several modified theories of gravity like $f(R)$ gravity,
Brans-Dicke gravity, string inspired gravity and so on. In any case, exact
solutions are found out.
