limit. Then, we obtain the nonvanishing Higgs VEV $v=\sqrt{2m/\lambda}$. It follows that the equation (\[deru\]) results in $\hat{\theta}_1 =0$ or $\theta_1 =0$ mod $2\pi$. This result is also derived from Eq.(\[eqevendt\]) by taking the limit $\bar{m}\rightarrow \infty$. For these values of the order parameters, the equation we have to solve becomes the same equation as the one that produces the Hosotani phase for the case $N=$ odd, but in the present case, $N$ is replaced by $N-1$ ($=$ odd) with the nonvanishing $\bar{v}$. Hence, we finally arrive at the solution (\[higgsd\]).

Our task is now to solve the equation (\[eqevendt\]) for finite sizes of $S^1$ and to confirm the phase structure for the case $N=$ even depicted in the Fig.3. The coexisting phase is given by $${\rm type~~III}~~\cdots~~\left\{\begin{array}{l}
gL\vev{A_y}={\rm diag}\left(\theta_c, {N\over{N-1}}\pi
-{\theta_c\over{N-1}},\cdots,{N\over{N-1}}\pi-{\theta_c\over{N-1}},
{{-\theta_c}\over{N-1}}-{{N(N-2)}\over{N-1}}\pi\right),\\[0.3cm]
\vev{\Phi}={1\over\sqrt{2}}(\sqrt{2\over\lambda}v,0,\cdots, 0)^T,
\end{array}\right.
\label{coexistd}$$ where $\vbar =\sqrt{{2\over\lambda}(\mbar^2-(2\pi-\theta_c)^2)}$ and $\theta_c$ is the solution for the coexisting phase (see below). The $SU(N)$ gauge symmetry is broken to $SU(N-1)\times U(1)$ by the Wilson line and the Higgs VEV breaks the $U(1)$, so that the residual gauge symmetry is $SU(N-1)$ for the vacuum configuration (\[coexistd\]).

In order to confirm that the phase structure is actually given by the Fig.3, it is 