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A prolate $\gamma$-rigid version of the Bohr-Mottelson Hamiltonian with aquartic anharmonic oscillator potential in $\beta$ collective shape variable isused to describe the spectra for a variety of vibrational-like nuclei.Speculating the exact separation between the two Euler angles and the $\beta$variable, one arrives to a differential Schr\"{o}dinger equation with a quarticanharmonic oscillator potential and a centrifugal-like barrier. Thecorresponding eigenvalue is approximated by an analytical formula dependingonly on a single parameter up to an overall scaling factor. The applicabilityof the model is discussed in connection to the existence interval of the freeparameter which is limited by the accuracy of the approximation and bycomparison to the predictions of the related $X(3)$ and $X(3)$-$\beta^{2}$models. The model is applied to qualitatively describe the spectra for ninenuclei which exhibit near vibrational features.

Quartic oscillator potential in the {\gamma}-rigid regime of the collective geometrical model