The R2-curvature

Question

Let $(M,g)$ be a riemannian manifold with riemannian curavature $R \in \Lambda^2 (TM)\otimes End(TM)$, then I define the $R_2$-curvature:

$$R_2 = R(e_i,e_j)R(e_i,e_j)$$

with an orthonormal basis $(e_i)$. $R_2$ is symmetric and negativ.

We can then define the Einstein equations for $R_2$-curvature:

$$ R_2 = - Id$$

I define:

$$r_2 (X,Y)= g(R_2 (X),Y)$$

The $r_2$-curvature flow is then:

$$\frac{\partial g}{\partial t}=r_2$$

Can we have solutions for the $r_2$ flow?