  Based on operator identities and their formal adjoints, we derive two
symmetry operators for the linearized Einstein operator on vacuum backgrounds
of Petrov type D and in particular the Kerr spacetime. One of them is of
differential order four and coincides with a result of Cohen and Kegeles. The
other one is a new operator of differential order six. The corresponding
operator identities are based on the Teukolsky equation and the
Teukolsky-Starobinski identities, respectively. The method applies to other
field equations as well, which is illustrated with the Maxwell equation. The
resulting symmetry operators are connected to Hertz and Debye potentials as
well as to the separability of the Teukolsky equation for both Maxwell and
linearized gravity.
