An infinitesimal Rattray theorem

Let $X$ be a continuous vector field on the unit Euclidean sphere centered at the origin such that $X(-a)=-X(a)$. It is proved that there is an orthonormal basis in the space such that for any two vectors $a$ and $b$ in the basis we have $X(a)\cdot b+a\cdot X(b)=0$. Bibl. – 1 title.