  In the context of metric f(R) 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.
