Invariant solutions and Noether symmetries in Hybrid Gravity
  Symmetries play a crucial role in physics and, in particular, the Noether
symmetries are a useful tool both to select models motivated at a fundamental
level, and to find exact solutions for specific Lagrangians. In this work, we
consider the application of point symmetries in the recently proposed
metric-Palatini Hybrid Gravity in order to select the $f({\cal R})$ functional
form and to find analytical solutions for the field equations and for the
related Wheeler-DeWitt (WDW) equation. We show that, in order to find out
integrable $f({\cal R})$ models, conformal transformations in the Lagrangians
are extremely useful. In this context, we explore two conformal transformations
of the forms $d\tau=N(a) dt$ and $d\tau=N(\phi) dt$. For the former conformal
transformation, we found two cases of $f({\cal R})$ functions where the field
equations admit Noether symmetries. In the second case, the Lagrangian reduces
to a Brans-Dicke-like theory with a general coupling function. For each case,
it is possible to transform the field equations by using normal coordinates to
simplify the dynamical system and to obtain exact solutions. Furthermore, we
perform quantization and derive the WDW equation for the minisuperspace model.
The Lie point symmetries for the WDW equation are determined and used to find
invariant solutions.
