= Solution
Let $u$ be the <tangent vector> to a <geodesic> with <affine parameter> $\lambda$. Its <geodesic equation> is $\nabla_uu=0$. For a <Killing vector field> $\xi$, <metric compatibility> gives
$$
\frac{d}{d\lambda}(\xi_au^a)=u^au^b\nabla_b\xi_a+\xi_a u^b\nabla_bu^a=u^au^b\nabla_{(a}\xi_{b)}=0.
$$
Thus
$$
\boxed{\xi_au^a\text{ is constant along every affinely parametrized geodesic}.}
$$
This <geodesic conserved quantity from a Killing vector> is independent of whether the <geodesic> is timelike, spacelike or null. For a <timelike geodesic> one can use <proper time>; for a <null geodesic> choose an <affine parameter>. An arbitrary nonaffine tangent rescales the displayed quantity and need not make it constant.
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