= 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:
$$
\dot x\big|_{(S,C)}=-\tfrac12 C^2S+2(SC/4)C=0,
$$
$$
\dot z\big|_{(S,C)}=-2(SC/4)S-\tfrac12(C^2+1)C+C
=\tfrac C2(1-S^2-C^2)=0.
$$
Taking $y=0$ makes its derivative zero as well. Hence \b[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 $C\ne0$, $D=C^2(C^2+1)+S^2C^2=2C^2$, and the expressions in (d) reduce to $x_*=S$, $z_*=C$, 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.
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