vacua of the theory. As we have already mentioned in section four, the $SU(2)$ $n=2$ model has four vacua. It turns out that the solutions  interpolate between two different vacua only for $\rho=0$, in which case $U=W=0$. This implies that all solitons of the $SU(2)$ $n=2$ model lie on the maximal torus. The same applies for all solitons of the $SU(2)$ $n=2$ model with $\kappa=1$.

To explain why there are no solitons of the $SU(2)$ $n=2$ model that lie in the complement of the maximal torus of $SU(2)$, let us study the vacuum structure of this model in more detail. A linearisation of the equations  around the vacua   reveals that $({\theta_0\over2}, 0)$ and $({-\theta_0+2\pi \over2}, \pi)$ are sources, and $({\theta_0+\pi
\over2},\pi)$ and $({-\theta_0+ \pi \over2}, 0)$ are saddles (for the Bogomol’nyi equations with the plus sign). Solving the equations  with $n=2$ for $\theta$ and $\phi$ by setting $\phi=0,\pi$, we find precisely the solitons of the $SU(2)$ $n=2$ model that lie on the maximal torus of $SU(2)$ and therefore are embeddings of the $SO(2)$ solitons. These solitons interpolate between the source ($\phi=0$) and the saddle ($\phi=0$), the saddle ($\phi=0$) and the source ($\phi=\pi$), the source ($\phi=\pi$) and the saddle ($\phi=\pi$), and 