The generalization of the Killing spinor equation to nonconstant Killing function $\lambda$ has been worked out in

H.-B. Rademacher, Generalized Killing spinors with imaginary Killing function and conformal Killing fields, Lecture Notes in Math. 1481 (Springer, Berlin, 1991).

A nonconstant $\lambda$ is only possible if its real part is identically zero.

A more recent paper along these lines is Complex Generalized Killing Spinors on Riemannian Spin manifolds (2013).

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.

The generalization of the Killing spinor equation to nonconstant Killing function $\lambda$ has been worked out in

H.-B. Rademacher, Generalized Killing spinors with imaginary Killing function and conformal Killing fields, Lecture Notes in Math. 1481 (Springer, Berlin, 1991).

A nonconstant $\lambda$ is only possible if its real part is identically zero.

A more recent paper along these lines is Complex Generalized Killing Spinors on Riemannian Spin manifolds (2013).

The generalization of the Killing spinor equation to nonconstant Killing function $\lambda$ has been worked out in

H.-B. Rademacher, Generalized Killing spinors with imaginary Killing function and conformal Killing fields, Lecture Notes in Math. 1481 (Springer, Berlin, 1991).

A nonconstant $\lambda$ is only possible if its real part is identically zero.

A more recent paper along these lines is Complex Generalized Killing Spinors on Riemannian Spin manifolds (2013).