 Solution

Write $S=\sin {\theta }_d$, $C=\cos {\theta }_d$. The chosen feedback parameters give $q=-C$ and $\alpha =SC/4$. Substitute the desired state directly into the dynamics, rather than dividing by a determinant that can vanish:Taking $y=0$ makes its derivative zero as well. Hence the prescribed state is stationary for every target angle, including the equator. It is pure because $S^2+C^2=1$ in this Pauli normalization.

For $C\ne 0$, $D=C^2(C^2+1)+S^2{C}^2=2C^2$, and the expressions in (d) reduce to $x_{\ast }=S$, $z_{\ast }=C$, confirming uniqueness. For $C=0$, both $q$ and $\alpha$ vanish and $D=0$. The desired equatorial state is still stationary by direct substitution, but belongs to the whole stationary segment $(x,0)$; the singular formula cannot be used to claim unique stabilization.