TITLE: Vector Fields in a Riemannian Manifold
QUESTION [6 upvotes]: Suppose $(M,g)$ is a Riemannian manifold. 
Is there a way to classify manifolds where there exists a vector field that commutes with the laplace beltrami operator?
Thanks

REPLY [8 votes]: I give a geometric explanation of the calculations of Willie, which simultaneously elaborates the suggestion of Deane. 
The flow of a vector field commuting with Laplacian preserves the Laplacian  and therefore its symbol which is the inverse of the metric (i.e., $g^{ij}$). Then, the flow of the vector field preserves the metric and the vector field is Killing 
A related question is 
is $\nabla \cdot ( c^2 \nabla)$ a Laplace-Beltrami operator?
Added later: Just have seen that  Tobias Fritz  in his comment which appeared before my answer already had the same explanation.