we must do some assumptions on the behavior of axial and non axial modes of deformation. We will assume that

-   Our choice of variables corresponds to independent degrees of freedom.

-   Non-axial vibrations are confined to their lowest stationary state.

-   An approximation similar to that of the X(5) model is valid for the differential equations of all non-axial amplitudes: [*i.e.*]{}, the differential equation in $\beta_2,\ \beta_3$ can be approximately decoupled from those concerning the other degrees of freedom.

Therefore, the complete wavefunction $\Psi$ of Eq. \[E:5.0\] can be factorized in three parts, as in Eq. 30 of I: $$\Psi=\Psi_0(\beta_2,\beta_3)\  \Psi_1\ Y_{JM}(\hat{\Omega})
\label{E:5.0}$$ where the function $\Psi_1$ depends on the deformation variables different from $\beta_2,\ \beta_3$.

From the Eq. \[E:2.7\] we know that also the determinant $G$ is factorized in the same way. Then, the differential equation for $\beta_2 ,\ \beta_3$ takes the form\
$$\begin{aligned}
&&\hspace*{-4mm}
\left\{ G_0^{-1/2}\!\left[ \frac{\partial}{\partial 
\beta_2}\!\left( G_0^{1/2}
\!\frac{\partial}{\partial \beta_2 }\right)\!+\! 
\frac{\partial}{\partial \beta_3}\!\left( G_0^{1/2} 
\frac{\partial}{\partial \beta_3 }\right) \right] \right. \ \\
&&\hspace*{-4mm}\left. +\ \epsilon - V(\beta_2,
\beta_3) - \frac{J(J+1)}{3(\beta_2^2+2\beta_3^2)} 
\right\} \Psi (\beta_2, \beta_3) = 0 \nonumber
\label{E:5.1} \end{aligned}$$ This equation can be somewhat simplified with the substitution $$\begin{aligned}
\Psi_0(\beta_2,\beta_3) = g^{-1/2}\ \Phi (\beta_2,\beta_3)
\label{E:5.2}\end{aligned}$$ where $g\propto 