Let $(M,g)$ be a $n$-dimensional Riemannian manifold with smooth metric $g$. 


Proposition 3.2 in the paper ["The Boost Problem in General Relativity"](https://doi.org/10.1007/BF01213014) by O'Murchadha and Chistodoulou 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 $\text{Riem}$ and $\nabla\text{Riem}$ respectively. Does anyone know a proof of this? Any reference is appreciated.