  In the vector space of algebraic curvature operators we study the reaction
ODE $$\frac{dR}{dt} = R^2+R^{#}= Q(R)$$ which is associated to the evolution
equation of the Riemann curvature oper- ator along the Ricci flow. More
precisely, we analyze the stability of a special class of zeros of this ODE up
to suitable normalization. In particular, we show that the ODE is unstable near
the curvature operators of the Riemannian product spaces $M \times
\mathbb{R}^k, \ k \geq 0$ where $M$ is an Einstein (locally) symmetric space of
compact type and not a spherical space form when $k = 0.$
