FLRW metric $f(R)$ cosmology with a perfect fluid by generating integrals of motion

In the context of metric f® gravity, we consider a FLRW space-time, filled with a perfect fluid described by a barotropic equation of state (p = \gamma \rho). We give the equivalent mini-superspace description and use the reparametrization invariance of the resulting Lagrangian to work in the equivalent constant potential description. At that point, we restrict our analysis to those models for which the ensuing scaled mini-superspace is maximally symmetric. Those models exhibit the maximum number of autonomous integrals of motion linear in the momenta, which are constructed by the Killing vectors of the respective mini-supermetric. The integrals of motion are used to analytically solve the equations of the corresponding models. Finally, a brief description of the properties of the resulting Hubble parameters is given.