A flow for spinorial connections

Question

For a spin manifold $(M,g)$, I define a flow for connections over spinors:

$$\frac{\partial \nabla_X }{\partial t}(\psi)= e_i .R_{\nabla}(e_i,X)(\psi)$$

where $R_{\nabla}$ is the curvature of the connection $\nabla$.

Can we have solutions for the spinorial flow?