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

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