The Dirac-Killing spinors

Question

Let $(M,g)$ be a spin manifold. I define a Dirac-Killing spinor $\psi$ by the following equation:

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

where ${\cal D}$ is the Dirac operator, $\nabla$ is the spinorial connection and $\Delta$ is the Laplacian.

What is the space of Dirac-Killing spinors?