  In this note, we study symmetry of solutions of the elliptic equation
\begin{equation*} -\Delta _{\mathbb{S}^{2}}u+3=e^{2u}\ \ \hbox{on}\ \
\mathbb{S}^{2}, \end{equation*} that arises in the study of rigidity problem of
Hawking mass in general relativity. We provide various conditions under which
this equation has only constant solutions, and consequently imply the rigidity
of Hawking mass for stable constant mean curvature (CMC) sphere.
