Let $u$ be a smooth function on $\mathbb S^2$, and assume that for every killing vector field $V$ on $\mathbb S^2$.


$$\int\_{\mathbb S^2} V(u) x\_j dS=0\text{,}\forall j=1,2,3$$


Is $u$ necessarily constant? 



Killing vector fields on sphere
The answer is "no".


Choose a basis $V\_1, V\_2, V\_3$ of Killing fields.


Note that
$$\int\limits\_{\mathbb S^2} V\_iu\cdot x\_j\cdot d\,\mathrm{area}
=
-\int\limits\_{\mathbb S^2} u\cdot V\_ix\_j\cdot d\,\mathrm{area}$$
Threfore you can take any $u$ which is orthogonal to each of 9 functions $s\_{i,j}=V\_ix\_j$. 


