This is a simple question, but I can't find explicit discussion in literatures that I can find. Real/imaginary Killing spinor equation
\begin{equation} \nabla_\mu \psi = \lambda \gamma_\mu \psi \end{equation} where $\lambda$ is constant is a well known subject. But I wonder why the case where $\lambda$ is non-constant receives no discussion?
One reason I can think of is, one may perform Weyl rescaling to bring $\lambda$ to constant, so the non-constant case is not interesting.
The freedom to vary $\lambda$ is severely constrained: For a nontrivial solution of the Killing spinor equation you need either a real and constant $\lambda$ or a function $\lambda$ with purely imaginary values [Lichnerowicz (1987)].
The generalization of the Killing spinor equation to nonconstant imaginary Killing function $\lambda$ has been studied in
H.-B. Rademacher, Generalized Killing spinors with imaginary Killing function and conformal Killing fields, Lecture Notes in Math. 1481 (Springer, Berlin, 1991).
See also Eigenvalues of the Dirac operator, Twistors and Killing Spinors on Riemannian Manifolds.
