theory. This will allow us to find the pairs of vacua that are connected by solitons. The results of this analysis for $n=3,4,5$ have being confirmed with a numerical computation (see Figs. \[1,2,3\]).

It is sufficient to examine the linearisation properties of the equations  in the interval $0\leq
\theta <{\pi\over2}$. This is because the equations   are invariant under the discrete symmetries $$(\theta, \phi, x)\rightarrow (\pi-\theta, \pi-\phi, x)
\eqn\nthreed$$ for $n$ even, and $$(\theta, \phi, x)\rightarrow (\pi-\theta, \phi, -x)
\eqn\nfourd$$ for $n$ odd, which can be used to extend the analysis to the whole range of $\theta$, ($\theta\in [0,\pi)$). Another property of the equations  for $n$ [*even*]{} is that, apart from the vacua of the theory, there is another fixed point at $(\theta, \phi)=({\pi\over2}, {\pi\over2})$.

Next we order the vacua $(\theta,\phi)$ in  with respect to the value of $\theta$, and denote these values by $\theta_i$, $i=1,\dots, 
2n$, [i.e.]{} $\theta_1<\dots
\theta_i<\theta_{i+1}\dots<\theta_{2n}$. For $\theta$ in $[0,\pi/2)$, a linearisation of  with the plus sign reveals that the vacua $\theta_{4q+1}$ are sources, the vacua $\theta_{4q+3}$ are sinks and the rest are saddle points. Observe that the sources $\theta_{4q+1}$ and the saddles $\theta_{4q+2}$ lie at the $\phi=0$ semicircle, while the sinks $\theta_{4q+3}$ and the saddles $\theta_{4q+4}$ 