= Solution
Choose a local transverse three-dimensional section of the <null geodesic congruence> and a future unit <timelike vector> $T$ on it. Orient $U$ to the future and put $E=-U\cdot T>0$. On that section define a <parallel auxiliary null vector> by the initial value
$$
N^a=\frac{T^a}{E}-\frac{U^a}{2E^2}.
$$
Since $T^2=-1$, $U^2=0$, and $U\cdot T=-E$, direct contraction gives $U\cdot N=-1$ and $N^2=-E^{-2}+E^{-2}=0$.
Extend $N$ along each generator by <parallel transport>, solving $U^b\nabla_bN^a=0$ with those initial values. The <geodesic equation> and compatibility of the <Levi-Civita connection> imply
$$
U\cdot\nabla(N^2)=0,\qquad U\cdot\nabla(U\cdot N)=0.
$$
Therefore \b[$N^2=0$, $U\cdot N=-1$, and $\nabla_U N=0$ hold throughout the local congruence]. Smooth initial data and the transport equation give a smooth field up to the breakdown of the congruence at caustics. An arbitrary pointwise choice of $T$ away from the initial section would not automatically have this transport property.
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