Classification of non-CSC extremal Kähler metrics on K-surfaces \(S^2_{\{\alpha \}}\) and \(S^2_{\{\alpha ,\beta \}}\)

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