  We prove that the $\mathrm{Sp}(1)$-Seiberg-Witten equation over a closed
hyperbolic $3$-manifold ${\mathbb H}^3/\Gamma$ always admits a canonical
irreducible solution induced by the hyperbolic metric. We also prove that the
Zariski tangent space of the moduli space at this canonical solution is same as
the Zariski tangent space of the moduli space of locally conformally flat
structures at the hyperbolic metric. This space is again same as the space of
trace-free Codazzi tensors and carries an injection to $H^1(\Gamma,\mathbb
R^{3,1})$, the first group cohomology of the $\Gamma$-module $\mathbb R^{1,3}$.
In particular, if $H^1(\Gamma,\mathbb R^{3,1})=0$ then the canonical
irreducible solution is infinitesimally rigid. We also prove that the
$\mathrm{Sp}(1)$-Seiberg-Witten equation over $S^1\times \Sigma$ has no
irreducible solutions and the moduli space of reducible solutions is same as
the moduli space of flat $\mathrm{SU}(2)$-connections.
