### Critical Points. Discussion. We use Equation (34) with an aim to investigate the critical points for the system of interest. We consider two distinct cases:  
**Case 1. **\(z=0\) and \(r=\rho\). Then Equations (34) modify appropriately  
\[ \begin{split}\det\mathsf{H}&=\frac{3\
u^{2}}{\rho^ {4}}\bigg{(}\lambda^{2}-\frac{1}{\rho^{3}}\bigg{)}+\lambda^{2}\bigg{(}1+\frac{ 2}{\rho^{3}}\bigg{)}-\frac{1}{\rho^{3}}\bigg{(}1+\frac{2}{\rho^{3}}\bigg{)}\,\\ \text{Tr}\,\mathsf{H}&=1+\lambda^{2}+\frac{3\
u^{2}}{ \rho^{4}}+\frac{1}{\rho^{3}}\.\end{split}  \tag{35}\]  
We discriminate among the following sub-cases:  
_Subcase (i):_\(\
u=0,\ z=0,\ \rho=1\). We obtain a system of equations as follows  
\[ \begin{split}\text{Tr}\,\mathsf{H}&=2+\lambda^{2}>0 \,\\ \det\mathsf{H}&=3\Big{(}\lambda^{2}-1\Big{)}\.\end{split}  \tag{36}\]  
Moreover, we infer a table (see Table 1) that describes the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) of the Hessian matrix:  
Thus, by investigating the signs of the Hessian matrix eigenvalues we can discriminate between critical points, minimum points, saddle points and degeneracy. Only if \(\lambda=1\) the critical point is degenerate. When the determinant of the Hessian matrix of the potential \(\det\mathsf{H}\
eq 0\) the system is non-degenerate. The system is degenerate if \(\det\mathsf{H}=0\). _Subcase (ii):_\(z=0,\ \
u\
eq 0\). \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(\mathbf{\lambda_{1}}\) & \(\mathbf{\lambda_{2}}\) & **Critical Point** \\ \hline \(0<\lambda<1\) & \(>0\) & \(>0\) & Minimum \\ \hline \(\lambda=1\) & \(>0\) & \(0\) & Degeneracy \\ \hline \(\lambda>1\) & \(>0\) & \(<\)0 & Saddle point \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Hessian matrix eigenvalues and associated critical points. The derivatives of a smooth function must be continuous. We now turn back to Equation (29). Then \(\
u^{2}=\rho\big{(}\rho^{3}-1\big{)}\) and  
\[\det\mathsf{H}=\bigg{(}4-\frac{1}{\rho^{3}}\bigg{)}\bigg{(}\lambda^{2}-\frac{1} {\rho^{3}}\bigg{)}. \tag{37}\]  
We seek for degenerate critical points (characterized by \(\det\mathsf{H}=0\)). We infer \(\rho=\lambda^{-2/3}\) or \(\rho=4^{-3}\), which involves two distinct sub-subcases:  
1. \(\rho=\lambda^{-2/3}\). We return to Equation (27) and infer \[\lambda^{-8/3}-\lambda^{-2/3}-\
u^{2}=0\.\]
2. \(\rho^{4}\geq\rho\,\ \rho\geq 1\). In such a situation we encounter a point of minimum when \(\rho>\lambda^{-2/3}\), while the case \(\rho<\lambda^{-2/3}\) implies a saddle point. 3. \(r=\lambda^{-4/3}\,\ z^{2}=\lambda^{-4/3}-\rho^{2}\). In this particular situation, after doing the math Equation (34) can be recast into  
\[ \begin{split}\det\mathsf{H}=12\lambda^{2}\Big{(}1-\lambda^{2} \Big{)}\Big{(}1-\lambda^{4/3}\rho^{2}\Big{)}\,\\ \operatorname{Tr}\mathsf{H}=4-\lambda^{2}>0\,\end{split}  \tag{38}\]  
with \(0\leq\lambda^{2}\leq 1\). We differentiate among several sub-cases as follows:  
_Subcase (i)_: \(\
u=0\,\ \lambda^{2}=1\). We further infer \(z=\pm\sqrt{\lambda^{-4/3}-\rho^{2}}\), with \(\rho^{2}\in\left[-\lambda^{2/3},\lambda^{2/3}\right]\), \(\rho\leq-\lambda^{-8/3}\). Then \(\operatorname{Tr}\mathsf{H}=3\) and \(\det\mathsf{H}=0\), which characterizes a degenerate critical point. _Subcase (ii)_: \(\
u\
eq 0\,\ \lambda^{2}=0\). We are in the case of a degenerate critical point, with \(\rho=\sqrt{|\
u|}\). In this particular case \(\det\mathsf{H}=0\) and  
\[\
u^{2}=\lambda^{-8/3}-\lambda^{-2/3}\.\]  
_Subcase (iii)_: \(0\leq\lambda^{2}<1\,\ \
u\
eq 0\). The critical point is a point of minimum, as \(\det\mathsf{H}>0\):  
\[\rho_{0}=\frac{\sqrt{|\
u|}}{\sqrt[4]{1-\lambda^{2}}}\,\ 1-\lambda^{2}>\
u^{2} \lambda^{4/3}\,\]  
\[z_{1,2}=\sqrt{\lambda^{-4/3}-\frac{\
u}{\sqrt{1-\lambda^{2}}}}\.\]  
_Subcase (iv)_: \(0<\lambda^{2}<1\,\
u=0\). Then \(\rho=0\) and \(\det\mathsf{H}=0\), which indicates a point of minimum characterized by \(z_{12}=\pm\lambda^{-2/3}\). _Subcase (v)_: Case \(\
u=0\,\lambda=0\). In such case we infer \(\rho=0\,z=0\). We are in the case of a degenerate critical point as \((\det\mathsf{H}=0)\).