  Let (M,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,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.
