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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 ofsolutions with generalized Nahm pole singularities along a simple divisor (a"knot") in $S^1\times\Sigma \times \{0\}$.

Classification of Nahm pole solutions of the Kapustin-Witten equations on $S^1\times \Sigma\times \mathbb{R}^+$