The extended Bogomolny equations on ${\mathbb{R}}^2\times {\mathbb{R}}^{+}$ I: Nilpotent Higgs field

We study the extended Bogomolny equations with gauge group $SU(2)$ on ${\mathbb{R}}^2\times {\mathbb{R}}^{+}$ with generalized Nahm pole boundary conditions and nilpotent Higgs field. We completely classify solutions by relating them to certain holomorphic data through a Kobayashi-Hitchin correspondence.