  We commonly refer to an extremal K\"{a}hler metric with finitely many
singularities on a compact
  Riemann surface as an HCMU (Hessian of the Curvature of the Metric is
Umbilical) metric. In this study, we specifically classify non-CSC HCMU metrics
on the K-surfaces $S^2_{\{\alpha\}}$ and $S^2_{\{\alpha,\beta\}}$.
