  In this note, we classify all solutions to the $\mathrm{SU(n)}$
Kapustin-Witten equations on $S^1\times\Sigma \times \mathbb{R}^+$, where
$\Sigma$ is a compact Riemann surface, with Nahm pole singularity at
$S^1\times\Sigma \times \{0\}$. We provide a similar classification of
solutions with generalized Nahm pole singularities along a simple divisor (a
"knot") in $S^1\times\Sigma \times \{0\}$.
