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Based on the competition between $\gamma$-stable and $\gamma$-rigidcollective motions mediated by a rigidity parameter, a two-parameter exactlyseparable version of the Bohr Hamiltonian is proposed. The $\gamma$-stable partof the Hamiltonian is restricted to stiff oscillations around the $\gamma$value of the rigid motion. The separated potential for $\beta$ and $\gamma$shape variables is chosen such that in the lower limit of this parameter, themodel recovers exactly the ES-$X(5)$ model, while in the upper limit it tendsto the prolate $\gamma$-rigid solution $X(3)$. The combined effect of therigidity and stiffness parameters on the energy spectrum and wave function isduly investigated. Numerical results are given for few nuclei showing suchambiguous behaviour.

journal of physics. g, nuclear and particle physics

Conjunction of $\gamma$-rigid and $\gamma$-stable collective motion in  the critical point of the phase transition from spherical to deformed nuclear  shapes