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

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