The Einstein-Killing spinors

Question

Let $(M,g)$ be a spin manifold with Ricci curvature $Ric$. An Einstein-Killing spinor $\psi$ is defined by the following equation:

$$X.{\cal D}\psi =\mu Ric(X).\psi$$

where ${\cal D}$ is the Dirac operator and $X$ is a variable tangent vector, $\mu$ is a constant.

If the manifold $M$ is Einstein, then the spinor $\psi$ is a proper vector of the Dirac operator.

What is the space of Einstein-Killing spinors?

Answer 1

Taking the norm, we have that:

$$||Ric(X)||=\lambda ||X||$$

so that the manifold is Einstein and the spinor $\psi$ is a proper vector of the Dirac operator.

Another interesting equation would be:

$$\nabla_X {\cal D}  \psi = \mu Ric(X).\psi$$

with $\nabla$ the spinorial connection.