= Solution
The additional invariant is the squared <norm> of the affine tangent. <Metric compatibility> and the <geodesic equation> make it constant, and a photon has the <null condition>
$$
g_{\mu\nu}\dot x^\mu\dot x^\nu=0.
$$
On the equatorial plane this gives
$$
0=-fc^2\dot t^2+f^{-1}\dot r^2+r^2\dot\phi^2,
$$
and the <conserved quantities> of part (a) yield
$$
\boxed{
\dot r^2=c^2\varepsilon^2-\frac{\ell^2}{r^2}\left(1-\frac{2m}{r}\right).
}
$$
For a nonradial ray $\ell\ne0$, put $u=1/r$. Since
$$
\frac{du}{d\phi}
=\frac{-\dot r/r^2}{\ell/r^2}
=-\frac{\dot r}{\ell},
$$
the orbit equation becomes
$$
\boxed{
\left(\frac{du}{d\phi}\right)^2+u^2(1-2mu)
=C,\qquad C=\frac{c^2\varepsilon^2}{\ell^2}=\frac1{b^2}.
}
$$
Here $b=|\ell|/(c\varepsilon)$ is the positive <impact parameter> for a future-directed ray arriving from infinity. A strictly radial ray has $\ell=0$ and must be described by the radial equation rather than by $\phi$ as parameter.
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