Classification of non-CSC extremal Kähler metrics on K-surfaces $S_{\{\alpha \}}^2$ and $S_{\{\alpha ,\beta \}}^2$

We commonly refer to an extremal Kähler 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 $S_{\{\alpha \}}^2$ and $S_{\{\alpha ,\beta \}}^2$.