Trivial solution and symmetries of nontrivial solutions to a mean field equation

We consider the mean field equation $$ \frac {\alpha }{2}\varDelta _g u+e^u-1=0\quad \text {on } \mathbb {S}^2. $$ We show that under some technical conditions, $u$ has to be constantly zero for $ {1}/{3}\leq \alpha \lt 1$. In particular, this is the case if $u(x)=-u(-x)$ and $u$ is odd symmetric about a plane. In the cases $u(x)=-u(-x)$ with $ {1}/{3}\leq \alpha \lt 1$ and $u(x)=u(-x)$ with $ {1}/{4}\leq \alpha \lt 1$, we analyze the additional symmetries of the nontrivial solution in detail.