  In this paper we develop a Kobayashi-Hitchin type correspondence between
solutions of the extended Bogomolny equations on $\Sigma\times \RP$ with Nahm
pole singularity at $\Sigma \times \{0\}$ and the Hitchin component of the
stable $SL(2,\mathbb{R})$ Higgs bundle; this verifies a conjecture of Gaiotto
and Witten. We also develop a partial Kobayashi-Hitchin correspondence for
solutions with a knot singularity in this program, corresponding to the
non-Hitchin components in the moduli space of stable $SL(2,\mathbb{R})$ Higgs
bundles. We also prove existence and uniqueness of solutions with knot
singularities on $\mathbb{C}\ti\RP$.
