When we restrict the Dirichlet energy functional to the set of all unit vector fields on a compact Riemannian manifold $(M,g)$, then the critical points of this functional are satisfied in $\Delta_g X=||\nabla X||^2 X$ where $X$ is a unit vector field on $M$ and a critical point. ($\Delta _g$ is the Laplace-Beltrami operator).

Is it true that such a vector field is killing or conformal or affine-killing vector field? I mean how can i investigate the 1-parameter group of the vector field $X$ by the equation $\Delta_g X=||\nabla X||^2 X$?

**Update:** Thanks to Peter's introduced reference. I found out a killing vector field $X$ is a critical point if and only if $Ric(X,V)=0$ for all vector fields $V \in X^{\perp}$.