$SU(N)$ gauge symmetry is broken to $SU(N-1)\times U(1)$ by the Wilson line and the $U(1)$ symmetry is broken by the Higgs VEV, so that the residual gauge symmetry is $SU(N-1)$. Likewise, in the coexisting phase, the residual gauge symmetry is $SU(N-1)$.

It is important to note that each type of the vacuum configuration has the restricted region determined by the scale $\mbar$, in which the configuration is stable against small fluctuations. The region also depends on the parameter $t\equiv \lambda A=4\lambda N_F$. Let us quote relevant results from the appendix that are necessary to determine the phase structure of the model. The Hosotani phase (type I) is stable for $$0 < \mbar < {{N-1}\over N}\pi\equiv \mbar_2,
\label{regionu}$$ and the Higgs phase (type II) is stable when $\mbar$ satisfies $$\mbar > \left({{2N-3}\over{N-1}}{t\over{24}}\right)^{\half}\equiv
\mbar_3.
\label{regiond}$$ In the coexisting phase (type III), $\theta_1^-(\mbar)$ must satisfy the reality condition $(\theta_1^-({\mbar}))^*=\theta_1^-(\mbar) $ and $$0 \leq \theta_1^-({\mbar})\leq {{N-1}\over N}\pi .
\label{restrictodd}$$ These requirements on $\theta_1^-(\mbar)$ restrict the allowed region of the coexisting phase in the parameter space of $(\bar{m}, t)$. The analysis in the appendix shows that the coexisting phase lies in the region $$\begin{aligned}
\mbar_2 \leq \mbar \le \mbar_3 && \quad{\rm for}\quad
t\ge 48\pi^2\frac{(N-1)^3}{N(N^2-3)} ,
