,
\eqn\onecnew
$$
where $01$, the model does not have any supersymmetric vacua and
therefore supersymmetry is spontaneously
broken. For $01$, the supersymmetric vacua of the
$SO(2)$ model are
$$
\theta[\tilde \ell, \pm]={1\over n}\big(\pm\theta_0+ {1\mp 1\over 2} 
\pi + 2\pi \tilde\ell\big) \ , 
\eqn\threec
$$
where $$ is an integer,  $00 ]{} $$ interpolate between the vacua $\theta[\ell, -]$ and $\theta[ \ell+1, +]$, for $0\leq\ell<n-1$, and the vacua $\theta[n-1, -]$ and $\theta[0, +]$ for $\ell=n-1$. We remark that the solutions  are continuous and differentiable at $x=0$. The solitons are those solutions in  and in  with the plus sign whereas the anti-solitons are those with the minus sign. We note that the $SO(2)$ solitons of  and  interpolate between all neighbouring pairs of vacua of the model. (ii) For $\kappa=0$, the equation  becomes that of the sine-Gordon theory. The solutions are $$\theta=\zeta\, {2\over n}\bigg({\rm arc}\tan\big[\exp\big(\pm m n 
(x-x_0)\big)\big]+\ell \pi\bigg)
\eqn\sixc$$ for $0\leq \ell< n$, $\ell\in \bZ$ and $\zeta=\pm$. (iii) For $\kappa=1$ and $n>1$ , the solutions that interpolate between the vacua $\theta[\ell]$ and $\theta[\ell+1]$, for $0\leq\ell<n-1$, and the vacua $\theta[n-1]$ and $\theta[0]$ for $\ell=n-1$, are $$\theta={2\over n}\bigg({\rm arc}\tan\big[\mp m n (x-x_0)\big]-
{\pi\over4}+\ell \pi\bigg)\ . 
\eqn\sixcc$$ These are $2n$ solutions, $n$ solitons and 