\eqn\qnew$$ where $0\leq \ell<n$, $\ell \in \bN$. For $\kappa=1$, the $n$ vacua of the theory (without counting multiplicities) can be found by setting $\theta_0=\pi/2$ in .

The equations  written in the angular parameterisation are $$\eqalign{
{d\over dx}\theta&=\pm m[-\kappa \cos \phi+\sin(n\theta)]
\cr
{d\over dx}\phi&=\pm m\kappa \cot\theta \sin\phi
\cr
{d\over dx}\psi&=0\ .}
\eqn\ninedaan$$ The third equation above implies that the angle $\psi$ is constant; $\psi$ is related to the ratio of $U, W$ as $\cot\psi=\lambda$. So it remains to solve the first two equations of  for $\theta$ and $\phi$ for the various values of $n$ and $\kappa$.

To find a class of solutions for the equations  above, we set $\phi$ to its vacuum values, [i.e.]{} $\phi=0,\pi$. This choice for $\phi$ solves the second of the equations , so it remains to solve the first for $\theta$. Since we have set $\phi=0,\pi$, the solutions lie on the maximal torus of $SU(2)$, and they are best described by extending the range of $\theta$ from $0\leq
\theta<\pi$ in  to $0\leq \theta<2\pi$. Then, the first equation in  is precisely the equation  that we have encountered in the $SO(2)$ model. Moreover, the vacua  of the $SU(2)$ model can be identified with the vacua  of the $SO(2)$ model 