  An exactly separable version of the Bohr Hamiltonian which combines the
$\gamma$-stable and $\gamma$-rigid axial vibration-rotation is used to describe
the collective properties of few neutron rich transitional nuclei. The coupling
between the two types of collective motion is managed through a rigidity
parameter which also influences the geometry of the shape-phase space. While
the $\gamma$-angular part of the problem associated to axially symmetric shapes
is treated within the small angles approximation and the stiff $\gamma$
oscillation hypothesis, the $\beta$ vibration is described by means of a
Davidson potential. The resulting model have three free parameters not counting
the scale and was successfully applied for the description of the collective
spectra for few heavier isotopes of Gd and Dy. In both cases a critical nucleus
was identified through a discontinuous behavior in respect to the rigidity
parameter and relevant experimental observables.
