A vector field $\zeta$ is conformal on a Riemannian manifold $(M,g)$ if $$\mathcal L_\zeta g=\rho g$$These vector fields have a well known geometrical interpretation. The flow of a conformal vector field consists of conformal transformations.

I want to enlarge this class of conformal vector fields as follows. A vector field $\zeta$ is $2-$conformal on a Riemannian manifold $(M,g)$ if $$\mathcal L_\zeta \mathcal L_\zeta g=\rho g$$ It is clear that any conformal vector field is $2-$conformal and the converse need not be true. T. Operea, B. Unal and me did the same think for $2-$Killing vector fields(see references blow).

**My question is: **

What could be the physical and geometric interpretation of a $2-$conformal vector field?

In fact, I want to understand the left-hand side, is it a double consequent dragging of $g$ or some thing else?

**References**

1. $2-$Killing vector fields on Riemannian manifolds (http://www.emis.de/journals/BJGA/v13n1/B13-1.htm)

2. $2-$Killing vector fields on warped product manifolds (http://www.worldscientific.com/doi/abs/10.1142/S0129167X15500652)