= Solution
Set $m=GM/c^2$ and $f(r)=1-2m/r$. A photon follows a <null geodesic>, with an <affine parameter> $\lambda$ satisfying
$$
\frac{d^2x^\mu}{d\lambda^2}
+\Gamma^\mu{}_{\alpha\beta}
\frac{dx^\alpha}{d\lambda}\frac{dx^\beta}{d\lambda}=0.
$$
The <Christoffel symbols> follow from the supplied <Schwarzschild spacetime> <metric tensor>. An efficient equivalent procedure is to use the <geodesic Lagrangian>
$$
\mathcal L=\frac12\left[
-fc^2\dot t^2+f^{-1}\dot r^2
+r^2\dot\theta^2+r^2\sin^2\theta\,\dot\phi^2
\right],
$$
where dots denote $d/d\lambda$. Its <Euler-Lagrange equations> give the affinely parametrized <geodesic equations>. The condition $\mathcal L=0$ is imposed after variation; setting the Lagrangian identically to zero before varying would discard the equations.
By <spherical symmetry> the ray's <angular momentum> fixes an orbital plane through the center. Rotate the angular coordinates so that plane is $\theta=\pi/2$. The $\theta$ <Euler-Lagrange equation> is
$$
\frac{d}{d\lambda}(r^2\dot\theta)
=r^2\sin\theta\cos\theta\,\dot\phi^2,
$$
so $\theta=\pi/2$, $\dot\theta=0$ is preserved once imposed initially. A radial ray has no unique orbital plane, but any plane through that radial line can be chosen.
The coordinates $t$ and $\phi$ are cyclic. Thus their canonical momenta obey
$$
\frac{d}{d\lambda}(-c^2f\dot t)=0,\qquad
\frac{d}{d\lambda}(r^2\dot\phi)=0.
$$
The two <conserved quantities> can be chosen as
$$
\boxed{\varepsilon=f\dot t,\qquad \ell=r^2\dot\phi.}
$$
These are the <geodesic conserved quantities from Killing vectors> $\partial_t$ and $\partial_\phi$. With the chosen parameter normalization the conserved energy is $E=c^2\varepsilon$ and axial <angular momentum> is $\ell$; multiplying the affine tangent by a constant multiplies both quantities by that constant.
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