In a Riemannian Manifold $(M,g)$ a vector field $X$ is said to be Killing vector field if $L_X g$=0 and is said to be conformal if $L_X g= fg$ for some smooth real function $f$ on $M$.
Also, the notion of bi-Killing vector field is defined (in which we have $L_X (L_X g)=0$. So it is natural to define bi-conformal vector field as $L_X (L_X g)=fg$. The second Lie derivative of such vector field is proportional to the metric tensor.
But, I need to find a vector field $X\in \mathcal{X}M $ in a Riemannian manifold $(M,g)$ such that the second Lie derivative be proportional to $X$, in fact
Is there exist a vector field $X$ such that $L_X (L_X g)=\lambda X^\flat \otimes X^\flat$?
Where, $\lambda$ is a real non-zero constant.
Any suggestion is highly appreciated.