Let  $X$  be  a  non-vanishing real analytic vector  field  on an open  manifold $M$. What  kind  of  obstructions would  appear when we search for  a Riemannian metric  on $M$ such that the space  of  harmonic  functions would  be  invariant under the derivation operator $f \mapsto X.f$? 
A harmonic function is a function $f$ which satisfy $\Delta_g (f)=0$ where $\Delta_g$ is the Laplace operator associated to the metric $g$.