  The eigenvalue equation associated to the Bohr-Mottelson Hamiltonian is
considered in the intrinsic reference frame and amended by replacing the
harmonic oscillator potential in the $\beta$ variable with a sextic oscillator
potential with centrifugal barrier plus a periodic potential for the $\gamma$
variable. After the separation of variables, the $\beta$ equation is
quasi-exactly solved, while the solutions for the $\gamma$ equation are just
the angular spheroidal functions. An anharmonic transition operator is used to
determine the reduced E2 transition probabilities. The formalism is
conventionally called the Sextic and Spheroidal Approach (SSA) and applied for
several X(5) candidate nuclei: $^{176,178,180,188,190}$Os, $^{150}$Nd,
$^{170}$W, $^{156}$Dy, $^{166,168}$Hf. The SSA predictions are in good
agreement with the experimental data of the mentioned nuclei. The comparison of
the SSA results with those yielded by other models, such as X(5) \cite{Iache9},
Infinite Square Well (ISW) \cite{Raduta}, and Davidson (D) like potential
\cite{Raduta} for the $\beta$, otherwise keeping the spheroidal functions for
the $\gamma$, and the Coherent State Model (CSM)
\cite{Rad1,Rad2,Rad3,Rad4,RaSa,Rad5} respectively, suggests that SSA represents
a good approach to describe nuclei achieving the critical point of the
U(5)$\rightarrow$SU(3) shape phase transition.
