Cone spherical metrics, defined on compact Riemann surfaces, are conformal metrics with constant curvature one and finitely many cone singularities. Such a metric is termed reducible if a developing map of the metric has monodromy in \(\mathrm{U(1)}\) , and irreducible otherwise. Utilizing the polystable extensions of two line bundles on a compact Riemann surface X with genus \(g_X>0\) , we establish the following three primary results concerning these metrics with cone angles in \(2\pi {\mathbb {Z}}_{>1}\) : (1) Given an effective divisor D with an odd degree surpassing \(2g_X\) on X, we find the existence of an effective divisor \(D'\) in the complete linear system |D| that can be represented by at least two distinct irreducible cone spherical metrics on X.
(2) For a generic effective divisor D with an even degree and \(\deg D\ge 6g_X-2\) on X, we can identify an arcwise connected Borel subset in |D| that demonstrates a Hausdorff dimension of no less than \(\big (\deg D-4g_{X}+2\big )\) . Within this subset, each divisor \(D'\) can be distinctly represented by a family of reducible metrics, defined by a single real parameter.
(3) For an effective divisor D with \(\deg D=2\) on an elliptic curve, we can identify a Borel subset in |D| that is arcwise connected and exhibits a Hausdorff dimension greater than one. Within this subset, each divisor \(D'\) can be distinctly represented by a family of reducible metrics, defined by a single real parameter.
