to $SU(2)$ models with $\kappa<0$. We remark that linearising the theory around the vacua , the mass-square-matrix and charge-matrix of the fundamental states of the $SU(2)$ model are $${\cal M}=m^2 \pmatrix{&1-\kappa^2\quad  &0\quad  &0\cr
                &0\quad &1\quad &0\cr
                &0\quad &0\quad &1}\ ,
\eqn\massmatone$$ and $${\cal Q}=\pmatrix{&0\quad &0\quad &0\cr
                &0\quad &0\quad &-2\kappa\cr
                &0\quad &2 \kappa\quad &0}\ ,
\eqn\chargematone$$ respectively. Clearly for $|\kappa|<1$, ${\cal M}>{m^2\over 4} {\cal Q} {\cal Q}^t$ and the fundamental states are not BPS states. However, for $|\kappa|=1$, ${\cal M}={m^2\over 4} {\cal Q} {\cal Q}^t$ and all the fundamental states become BPS with one of them massless.

The equations  in this parameterisation for the $SU(2)$ model are $$\eqalign{
{d\over dx}M\pm m [-\kappa N+(1-M^2) \tilde M_n]&=0
\cr
{d\over dx}N\pm m[\kappa M-NM\tilde M_n]&=0
\cr
{d\over dx}U\mp m U M \tilde M_n&=0
\cr
{d\over dx}W\mp m W M \tilde M_n&=0\ ,}
\eqn\fivedn$$ where $$\tilde M_n=\sum^{[{n-1\over2}]}_{i=0}\pmatrix{n\cr 2i+1} 
(-1)^{i} M^{n-2i-1} (1-M^2)^{i}\ .
\eqn\sixdn$$ There are three independent equations in  since the fourth one is implied from the restriction that $M, N, U, W$ lie on a 3-sphere. Moreover from the last two equations of , it is easy to see that $$U=\lambda W\ ,
\eqn\sevendn$$ where $\lambda$ is a real constant. So we only need to 