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" 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 $Ric$ and $\nabla Ric$ respectively. Does anyone know a proof of this? Any reference is appreciated.