  A class of generalized Galileon cosmological models, which can be described
by a point-like Lagrangian, is considered in order to utilize Noether's Theorem
to determine conservation laws for the field equations. In the
Friedmann-Lema\^itre-Robertson-Walker universe, the existence of a nontrivial
conservation law indicates the integrability of the field equations. Due to the
complexity of the latter, we apply the differential invariants approach in
order to construct special power-law solutions and study their stability.
