A scalar Ricci flow

Question

Let $(M,g_t)$ be a riemannian manifold with riemannian curvature $R(g)$ and scalar curvature $r(g)$, I define:

$$X_g=dr(g)^*$$

then I can define a flow over metrics:

$$\frac{\partial g}{\partial t}(X,Y)=R(g)(X,X_g,Y,X_g)$$

It is symmetric in $X,Y$.

Is this flow well defined? Have we singularities for the flow?