 Solution

The constant $u=1/(3M)$ satisfies $u=3M{u}^2$. It is the circular photon orbit at $r=3M$, the photon sphere. A perturbation $u=u_c+\eta$ satisfies ${\eta }^{\prime \prime }=\eta$ to first order, so perturbations grow exponentially with orbital angle: the circular null orbit is unstable. Substitution into the first integral gives $E^2/L^2=1/(27M^2)$.