The Killing flow of spinorial connections

Question

Let $(M,g)$ be a spin manifold with spinorial connections $\nabla$ on the spinors bundle. I define the Killing flow of spinorial connections by the following equation:

$$\frac{\partial \nabla_X}{\partial t} \psi= \mu X.\psi$$

Have we solutions for the Killing flow of spinorial connections?

Answer 1

The Killing flow of spinorial connections is in fact:

$$\nabla_X (t)=\nabla_X (0)+\mu t X$$