of $t$ is further classified into two cases, depending on the relative magnitude between $\mbar_5$ and $\mbar_3$:

(iii-a)  $24\pi^2 {{(N-1)^3(N^2-N-1)}\over {N^3 (2N-3)^2}}
< t \leq 24\pi^2{{(N-1)^3}\over {N^2(2N-3)}}$

In this case, the relative magnitude of $\bar{m}_i\ (i=1,\cdots,5)$ is given by $\bar{m}_1<\bar{m}_4\le\bar{m}_5\le\bar{m}_3<\bar{m}_2$ (see Fig.1). It immediately follows that the vacuum configuration is uniquely determined as $$\mbox {vacuum configuration}=\left\{\begin{array}{lll}
\mbox{Hosotani phase} &\mbox{for}& \mbar < \mbar_4,\\
\mbox{coexisting phase} &\mbox{for}& \mbar_4 < \mbar < \mbar_3,\\
\mbox{Higgs phase} & \mbox{for}& \mbar_3 < \mbar.
\end{array}\right.
\label{solt}$$ The vacuum structure is similar to the case (ii), and the phase transition at $\bar{m}=\bar{m}_3$ ($\bar{m}=\bar{m}_4$) is the second (first) order.

(iii-b)  $ t\leq 24\pi^2 {{(N-1)^3(N^2-N-1)}\over {N^3 (2N-3)^2}}$

In this case, we have $\mbar_1 \le \mbar_3 \le \bar{m}_5 < \bar{m}_2$ (see Fig.1). We observe that the Hosotani and coexisting phases overlap between $\mbar_1$ and $\mbar_3$. Let us recall that the difference of the potential energy between the Hosotani phase and the coexisting one, $\Delta\Vb$, is the monotonically increasing function with respect to $\bar{m}$, and we find that $$\Delta \Vb(\mbar=\mbar_3)={t\over{2\lambda}}
\left({{2N-3}\over{24(N-1)}}\right)^2
\left(t- 24\pi^2 {{(N-1)^3(N^2-N-1)}\over {N^3 (2N-3)^2}}\right) \leq 0,
\label{valu}$$ [*i.e.*]{} $\Vb_{Hosotani} \leq \Vb_{coexisting}$ for the parameter region of $t$ under consideration. This implies that there is no coexisting phase for this parameter 