  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 \textit{reducible} if a developing map of the metric has
monodromy in ${\rm U(1)}$, and \textit{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 $2\pi{\mathbb Z}_{>1}$:
  \begin{itemize} \item[(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 $X$.
  \item[(2)] For a generic effective divisor $D$ with an even degree and $\deg
D\geq 6g_X-2$ 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.
  \item[(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, showcasing a
Hausdorff dimension of one. Within this subset, each divisor $D'$ can be
distinctly represented by a family of reducible metrics, defined by a single
real parameter.
