  In this paper, we present a theoretical study of a conjonction of
$\gamma$-rigid and $\gamma$-stable collective motions in critical point
symmetries of the phase transitions from spherical to deformed shapes of nuclei
using exactly separable version of the Bohr Hamiltonian with
deformation-dependent mass term. The deformation-dependent mass is applied
simultaneously to $\gamma$-rigid and $\gamma$-stable parts of this famous
collective Hamiltonian. Moreover, the $\beta$ part of the problem is described
by means of Davidson potential, while the $\gamma$-angular part corresponding
to axially symmetric shapes is treated by a Harmonic Osillator potential. The
energy eigenvalues and normalized eigenfunctions of the problem are obtained in
compact forms by making use of the asymptotic iteration method. The combined
effect of the deformation-dependent mass and rigidity as well as harmonic
oscillator stiffness parameters on the energy spectrum and wave functions is
duly investigated. Also, the electric quadrupole transition ratios and energy
sprectrum of some $\gamma$-stable and prolate nuclei are calculated and
compared with the experimental data as well as with other theoretical models.
