2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/75832We study the basic structure of a HCMU metric in a K-Surface with prescribed singularities. When the underlying smooth surface is $S^2$, we prove the necessary condition given in [1] for the existence of HCMU metric is also sufficient.Differential Geometry53C21, 53A05text