minimize them separately. There are the two possible minimum for the field strength part and for the covariant derivative part: the trivial solution with $\phi$ and $\tilde{X}_A^\mu$, $\tilde{X}_B^\mu$ equal to the null fields and the solution with $\phi$, $X_A^\mu$, $X_B^\mu$ proportional to the identity. In each cases both the field strength part and the covariant derivative part disappear. For the potential parts we have: $$\begin{aligned}
V_A=V_B=\left(\frac{\chi_{-1}}{\chi_0}\right)^2 &&  \textrm{for} \ \phi=\tilde{X}^\mu_A=\tilde{X}^\mu_B=0 \\
V_A=(\alpha^2_\phi+4\frac{4\Omega^2}{1+\Omega^2}\alpha^2_A-\frac{\chi_{-1}}{\chi_0})^2,  &&  \textrm{for} \ \phi=\alpha_\phi \textbf{I}, \ \tilde{X}^\mu_A=\alpha_A \textbf{I}^\mu, \  \tilde{X}^\mu_B=\alpha_B \textbf{I}^\mu \\
V_B=(\alpha^2_\phi+4\frac{4\Omega^2}{1+\Omega^2}\alpha^2_B-\frac{\chi_{-1}}{\chi_0})^2 &&\end{aligned}$$ Referring to the second case and minimizing the potentials, the minimum seems to be for $$\phi_0=\frac{\chi_{-1}}{\chi_0}\cos\alpha \textbf{I},\ \tilde{X}^\mu_{A0}=\tilde{X}^\mu_{B0}= \frac{1}{2}\sqrt{\frac{\chi_{-1}}{\chi_0}}\sqrt{ \frac{1+\Omega^2}{4\Omega^2}} \sin\alpha \textbf{I}^\mu \label{Vacuum}$$ However, the previous position is not allowed because the identity does not belong to the algebra under consideration. In general the non-triviality of the vacuum makes very difficult to explicit the vacuum configuration of the system in [@Goursac1] A. de Goursac, J.C. Wallet, and R. Wulkenhaar, using the matrix base formalism, have found an expressions from vacuum solutions deriving them from the relevant solutions equations of motion. Although, the complexity of the vacuum configuration makes the perturbative approach very complicated, in order to conduct some investigation in the next section will 