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.



Killing constant in Killing spinor equation
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](http://link.springer.com/article/10.1007/BF00401162) (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](http://link.springer.com/chapter/10.1007/BFb0083642#page-1), Lecture Notes in Math. 1481 (Springer, Berlin, 1991).


See also [Eigenvalues of the Dirac operator, Twistors and Killing Spinors on Riemannian Manifolds](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.147.668&rep=rep1&type=pdf). 


