Suppose $u$ is a smooth function on $S^2$ and $$ X_1=x_2\frac{\partial}{\partial x_3}-x_3\frac{\partial}{\partial x_2}, X_2=x_3\frac{\partial}{\partial x_1}-x_1\frac{\partial}{\partial x_3},X_3=x_1\frac{\partial}{\partial x_2}-x_2\frac{\partial}{\partial x_1} $$ be conformal killing vector fields in $\mathbb{R}^3$. If $u$ satisfies $$ u(x_1, x_2,x_3)=u(|x_1|, |x_2|,|x_3|) $$ and $X_iu\equiv 0$ for some $i\in\{1,2,3\}$. Can we prove $u$ is axially symmetric, i.e., $u=u(x_i)$.
This problem arise from lemma 8 of the paper Uniqueness of the mean field equation and rigidity of Hawking Mass.
It seems a easy problem but I currently have some difficulty argue this through. Here $u$ is actually the sulotion of the following minimal field equation $$ -\alpha\Delta_{S^2}u+1=e^{2u}, $$ where $\alpha\in [\frac{1}{2},1)$. I think no additional condition is needed here.
Apprecaite any help!