  A prolate $\gamma$-rigid version of the Bohr-Mottelson Hamiltonian with a
quartic anharmonic oscillator potential in $\beta$ collective shape variable is
used 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 quartic
anharmonic oscillator potential and a centrifugal-like barrier. The
corresponding eigenvalue is approximated by an analytical formula depending
only on a single parameter up to an overall scaling factor. The applicability
of the model is discussed in connection to the existence interval of the free
parameter which is limited by the accuracy of the approximation and by
comparison to the predictions of the related $X(3)$ and $X(3)$-$\beta^{2}$
models. The model is applied to qualitatively describe the spectra for nine
nuclei which exhibit near vibrational features.
