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

Competing $\gamma$-rigid and $\gamma$-stable vibrations in neutron rich Gd and Dy isotopes