first study the case $N=$ odd.

$N=$ odd
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There are three types of possible vacuum configurations, as shown in the appendix, $$\begin{aligned}
{\rm type~~I}&\cdots&\left\{\begin{array}{l}
gL\vev{A_y}={\rm diag}\left({{N-1}\over N}\pi,
\cdots,{{N-1}\over N}\pi, -{{(N-1)^2}\over N}\pi\right),\\[0.3cm]
\vev{\Phi}={1\over\sqrt{2}}(0,\cdots, 0)^T,
\end{array}\right. \label{hosotani}\\[0.3cm]
{\rm type~~II}&\cdots&\left\{\begin{array}{l}
gL\vev{A_y}={\rm diag}\left(0, \pi, \pi, \cdots, -(N-2)\pi\right),\\[0.3cm]
\vev{\Phi}={1\over\sqrt{2}}(\sqrt{2\over\lambda}m,0,\cdots, 0)^T,
\end{array}\right.\label{higgs}\\[0.3cm]
{\rm type~~III}&\cdots&\left\{\begin{array}{l}
gL\vev{A_y}={\rm diag}\left(\theta_1^-, \pi-{\theta_1^-\over{N-1}},\cdots,
\pi-{\theta_1^-\over{N-1}},
-((N-2)\pi+{\theta_1^-\over{N-1}})\right),\\[0.3cm]
\vev{\Phi}={1\over\sqrt{2}}(v,\cdots, 0)^T.
\end{array}\right.\label{coexist}\end{aligned}$$ The type III solution depends on the scale $\mbar$, and accordingly, $\vbar=\sqrt{{2\over\lambda}(\mbar^2-(\theta_1^{-})^2)}$ does as well. The explicit form of $\theta_1^-(\mbar)$ is given by Eq.(\[cosol\]) in the appendix. As we stated before, we call the vacuum configuration corresponding to the type I, II and III solutions the Hosotani, Higgs and coexisting phases, respectively.

Given the vacuum configuration, the gauge symmetry in each phase is generated by the generators $T^a$ of $SU(N)$ which commute with the Wilson line, $$W\equiv {\cal P}{\rm exp}\left(ig\oint_{S^1}dy \vev{A_y}\right)
={\rm diag}\left(\e^{i\theta_1}, \e^{i\theta_2}, \cdots, \e^{i\theta_N}
\right)\quad
{\rm and} \quad T^a\vev{\Phi}=0.$$ Let us note that the phase $\theta_i$ is defined modulo $2\pi$. It is easy to observe that the $SU(N)$ gauge symmetry is not broken in the Hosotani phase because the Wilson line for the configuration (\[hosotani\]) is proportional to the identity matrix, $W={\rm exp}(i({{N-1}\over N}\pi)){\bf 1}_{N\times N}$. In the Higgs phase, the 