and the Higgs potential minimizing, thanks to the zero in Eq.(\[configadj\]). Then, one of the $U(1)$’s is broken by the Higgs VEV, so that the residual gauge symmetry is $U(1)^{N-2}$. It is important to note that in the model there is only one phase whose structure does not depend on the size of the extra dimension. This is a new feature that has not been observed in the phase structures obtained in the previous section. If $N=$ even $(\geq 4)$, there appears no zero component in Eq.(\[configadj\]). One needs to study the effective potential carefully in order to determine the vacuum structure of the model. In case of the $SU(2)$ gauge group with $N_F$ massless adjoint fermions, we can perform fully analytic calculations, and the phase structure is found to be very similar to the one obtained in the previous paper, though the residual gauge symmetry in the Hosotani phase is given by $U(1)$ in this case.

Let us next consider the adjoint Higgs instead of the fundamental Higgs. Both $\vev{A_y}$ and $\vev{\Phi}$ belong to the adjoint representation under the $SU(N)$ gauge group. The cross term corresponding to the third term in the effective potential (\[effpotu\]) is replaced by $$g^2{\rm 