  We consider the application of group invariant transformations in order to
constrain a flat isotropic and homogeneous cosmological model, containing of a
Brans-Dicke scalar field and a perfect fluid with a constant equation of state
parameter $w$, where the latter is not interacting with the scalar field in the
gravitational action integral. The requirement that the Wheeler-DeWitt equation
be invariant under one-parameter point transformations provides us with two
families of power-law potentials for the Brans-Dicke field, in which the powers
are functions of the Brans-Dicke parameter $\omega_{BD}$ and the parameter $w$.
The existence of the Lie symmetry in the Wheeler-DeWitt equation is equivalent
to the existence of a conserved quantity in field equations and with
oscillatory terms in the wavefunction of the universe. This enables us to solve
the field equations. For a specific value of the conserved quantity, we find a
closed-form solution for the Hubble factor, which is equivalent to a
cosmological model in general relativity containing two perfect fluids. This
provides us with different models for specific values of the parameters
$\omega_{BD},$ and $w$. Finally, the results hold for the specific case where
the Brans-Dicke parameter $\omega_{BD}$ is zero, that is, for the O'Hanlon
massive dilaton theory, and consequently for $f\left( R\right) $ gravity in the
metric formalism.
