  In this paper, we consider the following curvature equation $$\Delta u+{\rm
e}^u=4\pi\biggl((\theta_0-1)\delta_0+(\theta_1-1)\delta_1
+\sum_{j=1}^{n+m}\bigl(\theta_j'-1\bigr)\delta_{t_j}\biggr)\qquad \text{in}\
\mathbb R^2,$$ $$u(x)=-2(1+\theta_\infty)\ln|x|+O(1)\qquad \text{as} \
|x|\to\infty,$$
  where $\theta_0$, $\theta_1$, $\theta_\infty$, and $\theta_{j}'$ are positive
non-integers for $1\le j\le n$, while $\theta_{j}'\in\mathbb{N}_{\geq 2}$ are
integers for $n+1\le j\le n+m$. Geometrically, a solution $u$ gives rise to a
conical metric ${\rm d}s^2=\frac12 {\rm e}^u|{\rm d}x|^2$ of curvature $1$ on
the sphere, with conical singularities at $0$, $1$, $\infty$, and $t_j$, $1\le
j\le n+m$, with angles $2\pi\theta_0$, $2\pi\theta_1$, $2\pi\theta_\infty$, and
$2\pi\theta_{j}'$ at $0$, $1$, $\infty$, and $t_j$, respectively. The metric
${\rm d}s^2$ or the solution $u$ is called co-axial, which was introduced by
Mondello and Panov, if there is a developing map $h(x)$ of $u$ such that the
projective monodromy group is contained in the unit circle. The sufficient and
necessary conditions in terms of angles for the existence of such metrics were
obtained by Mondello-Panov (2016) and Eremenko (2020). In this paper, we fix
the angles and study the locations of the singularities $t_1,\dots,t_{n+m}$.
Let $A\subset\mathbb{C}^{n+m}$ be the set of those $(t_1,\dots,t_{n+m})$'s such
that a co-axial metric exists, among other things we prove that (i) If $m=1$,
i.e., there is only one integer $\theta_{n+1}'$ among $\theta_j'$, then $A$ is
a finite set. Moreover, for the case $n=0$, we obtain a sharp bound of the
cardinality of the set $A$. We apply a result due to Eremenko, Gabrielov, and
Tarasov (2016) and the monodromy of hypergeometric equations to obtain such a
bound. (ii) If $m\ge 2$, then $A$ is an algebraic set of dimension $\leq m-1$.
