= Solution
Write $f(r)=1-2\mu/r$. In the equatorial plane, the <geodesic Lagrangian> is $\mathcal L=\frac12(fc^2\dot t^2-f^{-1}\dot r^2-r^2\dot\phi^2)$. The <cyclic coordinates> $t$ and $\phi$ give the conserved quantities
$$
\boxed{f\dot t=k,\qquad r^2\dot\phi=h.}
$$
The squared tangent norm is constant along an affinely parametrized <geodesic>, since metric compatibility gives $d[g(\dot X,\dot X)]/d\tau=2g(D\dot X/d\tau,\dot X)=0$. Hence
$$
\boxed{fc^2\dot t^2-f^{-1}\dot r^2-r^2\dot\phi^2=E.}
$$
For a massive particle parametrized by <proper time>, $E=c^2$; a <photon> has a null tangent, so $E=0$. This $E$ denotes the tangent norm, not the photon's physical energy. A future-directed exterior photon has $k>0$, proportional to its positive <Killing energy>. Its affine parameter can be rescaled, so there is no mass-shell normalization fixing $k$ to a particular positive value.
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