convenient to consider intersections of two functions defined by $$\begin{aligned}
F(\theta_1)&\equiv &2(\mbar^2-(2\pi - \theta_1)^2)(2\pi -\theta_1),\\
G(\theta_1)& \equiv &{{-t}\over{12\pi^2}}
\left(\alpha\theta_1^3 -3\pi \alpha\theta_1^2 +\beta \pi^2\theta_1
+ \gamma \pi^3\right).
\label{fgeq}\end{aligned}$$ Let us note that $F(\theta_1)=G(\theta_1)$, in fact, reproduces Eq.(\[eqevendt\]) and $G(\theta_1)$ is dependent of $\bar{m}$. Since the intersections of the functions $F(\theta_1)$ and $G(\theta_1)$ have different behavior for $t > 48\pi^2 (N-1)/N$ and $t < 48\pi^2 (N-1)/N$, as discussed in the appendix, it is convenient to investigate separately the phase structure for each region of $t$ (see Figs.4 and 5).

(i)  $t> 48\pi^2 {{N-1}\over N}$

In this parameterization of $t$, there is one solution denoted by $\theta_c$ for $\bar{m} > \pi$, as shown in Fig.4. The solution satisfies the condition (\[restricteven\]) and is found to be stable, as discussed in the appendix. Since the Hosotani phase is unstable for $\bar{m}>\pi$, the coexisting phase must be the vacuum configuration for $\bar{m}>\pi$.

When the scale $\bar{m}$ approaches $\pi$, $\theta_c$ becomes closer to $\pi$ and is finally identical to $\pi$ at $\bar{m}=\pi$. This implies that the type III solution (coexisting phase) becomes identical to the type I solution (Hosotani phase). As the scale becomes smaller than $\pi$, the solution is outside of the required region (\[restricteven\]). 