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We study moduli spaces of solutions to the extended Bogomolny equations on$\Sigma \times \mathbb{R_{+,y}}$ with gauge group$\operatorname{SL}(2,\mathbb{C})$ satisfying the generalized Nahm pole boundarycondition as $y\to 0$ and limiting to complex flat connections as $y\to\infty$. Refining the Kobayashi-Hitchin correspondence of \cite{MH2}, weidentify these moduli spaces with certain holomorphic lagrangian sub-manifoldsinside the moduli space of Higgs bundles.

The moduli space of solutions to the Extended Bogomolny equations on $\Sigma \times \mathbb{R_+}$