Second Order Symmetries of the Conformal Laplacian

Let $(M,{\rm g})$ be an arbitrary pseudo-Riemannian manifold of dimension at least $3$. We determine the form of all the conformal symmetries of the conformal (or Yamabe) Laplacian on $(M,{\rm g})$, which are given by differential operators of second order. They are constructed from conformal Killing $2$-tensors satisfying a natural and conformally invariant condition. As a consequence, we get also the classification of the second order symmetries of the conformal Laplacian. Our results generalize the ones of Eastwood and Carter, which hold on conformally flat and Einstein manifolds respectively. We illustrate our results on two families of examples in dimension three.