  In general, geometries of Petrov type II do not admit symmetries in terms of
Killing vectors or spinors. We introduce a weaker form of Killing equations
which do admit solutions. In particular, there is an analog of the
Penrose-Walker Killing spinor. Some of its properties, including associated
conservation laws, are discussed. Perturbations of Petrov type II Einstein
geometries in terms of a complex scalar Debye potential yield complex solutions
to the linearized Einstein equations. The complex linearized Weyl tensor is
shown to be half Petrov type N. The remaining curvature component on the
algebraically special side is reduced to a first order differential operator
acting on the potential.
