Killing vector fields on sphere

Question

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?

Answer 1

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 $s_{i,j}=V_ix_j$.