as follows: $$\theta[\ell,\pm]=\cases{\bigg({1\over
n}(\pm\theta_0+{1\mp1\over2}\pi+2\pi  \ell),
0\bigg)\ , \qquad\qquad 0\leq \ell\leq \big[{n-1\over2}\big]
\cr
\bigg({1\over n}\big(\pm\theta_0+{1\mp1\over2}\pi+2\pi (\tilde
\ell+{1\over2})\big), \pi\bigg)\ ,\qquad \big[{n-1\over2}\big]<
\ell<n \ ,}
\eqn\vacthreenew$$ where $\tilde \ell=n-\ell-1$. Furthermore, as can be easily seen from , the solitons of the $SU(2)$ model that lie on the maximal torus are static. From this we conclude that the solitons of the $SU(2)$ model that lie on the maximal torus are embeddings of the solitons of the $SO(2)$ model found in the previous section with $$\theta\rightarrow \pmatrix{e^{i\theta}& \quad 0\cr 0& \quad e^{-i\theta}}\ ,
\eqn\fourteend$$ for the associated value of $n$. The class of solutions that we have described includes those for which $\kappa$ vanishes. This is because, if $\kappa=0$, the second equation in  implies that $\phi$ is constant and the asymptotic behaviour of the solitons requires that it should be set to its vacuum values.

Note from  and  that the definition of the topological charge of the $G$-models with $G$ semisimple is different from the definition of the topological charge of the $SO(2)$ models. The value of the topological charge $Q_T$ of any soliton of the $SU(2)$ model, or any $G$-model with $G$ semisimple, interpolating between the two vacua $(\theta,\phi)$ and $(\theta', \phi')$ is $Q_T=a(\theta',
