Let $(M,g)$ be a $n$-dimensional Riemannian manifold with smooth metric $g$.
A smooth vector field $X$ is conformal Killing if
$$\nabla_iX_j + \nabla_jX_i - \frac{2}{n} \text{div} X \, \,g_{ij} = 0$$
Proposition 3.2 in the paper "The Boost Problem in General Relativity" by Murchadha and Chistoudoulo claims that any conformal Killing field will satisfy the third order PDE: $$\nabla^3 X = A\cdot \nabla X + B \cdot X$$ where $A$ and $B$ are linear combination of $Ric$ and $\nabla Ric$ respectively. Does anyone know a proof of this? Any reference is appreciated.