Constraints and analytical solutions of $f(R)$ theories of gravity using
  Noether symmetries
  We perform a detailed study of the modified gravity $f(R)$ models in the
light of the basic geometrical symmetries, namely Lie and Noether point
symmetries, which serve to illustrate the phenomenological viability of the
modified gravity paradigm as a serious alternative to the traditional scalar
field approaches. In particular, we utilize a model-independent selection rule
based on first integrals, due to Noether symmetries of the equations of motion,
in order to identify the viability of $f(R)$ models in the context of flat FLRW
cosmologies. The Lie/Noether point symmetries are computed for six modified
gravity models that include also a cold dark matter component. As it is
expected, we confirm that all the proposed modified gravity models admit the
trivial first integral namely energy conservation. We find that only the
$f(R)=(R^{b}-2\Lambda)^{c}$ model, which generalizes the concordance $\Lambda$
cosmology, accommodates extra Lie/Noether point symmetries. For this $f(R)$
model the existence of non-trivial Noether (first) integrals can be used to
determine the integrability of the model. Indeed within this context we solve
the problem analytically and thus we provide for the first time the evolution
of the main cosmological functions such as the scale factor of the universe and
the Hubble expansion rate.
