= Solution
In the equatorial plane, the Killing symmetries give <conserved quantities> $E=(1-2M/r)\dot t$ and $L=r^2\dot\phi$. The null normalization gives
$$
\dot r^2=E^2-\frac{L^2}{r^2}\left(1-\frac{2M}{r}\right).
$$
For $L\ne0$, put $u=1/r$; then $du/d\phi=-\dot r/L$. Consequently
$$
(u')^2=(E/L)^2-u^2+2Mu^3.
$$
Differentiating gives $u''+u=3Mu^2$, initially where $u'\ne0$ and then at <turning points> by continuity. The circular solution follows directly from the radial equation as well. The ratio $|L/E|$ is the <impact parameter> of the asymptotic null orbit.
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