Recently a cubic Galileon cosmological model was derived by the assumption that the field equations are invariant under the action of point transformations. The cubic Galileon model admits a second conservation law which means that the field equations form an integrable system. The analysis of the critical points for this integrable model is the main subject of this work. To perform the analysis, we work on dimensionless variables that are different from those of the Hubble normalization. New critical points are derived, while the gravitational effects which follow from the cubic term are studied.

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

In this paper, we complement the analysis involving the “fundamental” solutions of the Klein–Gordon equation in de Sitter space–times given by Yagdjian and A. Galstian (Comm. Math. Phys. 285, 293 (2009); Discrete and Continuous Dynamical Systems S, 2(3), 483 (2009)). Using the symmetry generators, we classify and reduce the underlying equations and show how this process may lead to exact solutions by quadratures.