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 \(S^2_{\{\alpha \}}\) and \(S^2_{\{\alpha ,\beta \}}\).