potential is given by the configuration which gives the lowest energy. Following these steps, one can obtain the vacuum structure of the model.

The equation (\[deru\]) leads to $$\begin{aligned}
&&\vbar =0,\label{hvevu}\\
&{\rm or}&\nonumber\\
&&-\mbar^2+{\lambda\over 2}\vbar^2+\hat{\theta}_1^2=0.
\label{hvevd}\end{aligned}$$ For the first case (\[hvevu\]), the equations (\[derd\]) and (\[dert\]) are unified into an equation, $${A\over \pi^2}\sum_{n=1}^{\infty}{{-1}\over n^3}
\left(\sin(n\theta_k)+\sin(n\sum_{i=1}^{N-1}\theta_i)\right)=0,
\quad k=1, \cdots, N-1.
\label{eqhosotani}$$ We call the solution to this equation type I, and the solution describes the Hosotani phase. On the other hand, for the second case (\[hvevd\]) we solve the coupled equations, $$\begin{aligned}
{2\over\lambda}(\mbar^2 -\hat{\theta}_1^2)\hat{\theta}_1
+{A\over \pi^2}\sum_{n=1}^{\infty}
{{-1}\over n^3}\left(\sin(n\theta_1)+\sin(n\sum_{i=1}^{N-1}\theta_i)
\right)&=&0,\label{eqcoexistu}\\
{A\over \pi^2}\sum_{n=1}^{\infty}{{-1}\over n^3}
\left(\sin(n\theta_k)+\sin(n\sum_{i=1}^{N-1}\theta_i)\right)&=&0,
\quad k=2, \cdots, N-1.
\label{eqcoexistd}\end{aligned}$$ A solution to the equation is called type II or type III, depending on whether the solution has the scale dependence on the extra dimension or not. The type II (III) corresponds to the Higgs (coexisting) phase, whose vacuum expectation values are independent of (dependent on) the scale of the extra dimension. In order to avoid unnecessary complexity, details of calculations to solve these equations will be given in the appendix. We find that it is convenient to discuss the vacuum structure separately, depending on whether $N$ is odd or even. Let us 