= Solution
Locally write the <null hypersurface> as $u=0$, $du\ne0$, with normal $\ell_\mu=f\partial_\mu u$, $f\ne0$. A vector $V$ is tangent exactly when $V(u)=0$. Nullness gives $\ell(u)=\ell^2/f=0$ on the hypersurface. \b[Hence $\boxed{\ell\in T\mathcal N}$]: its normal direction lies inside its tangent space.
Put $n=du$ and $q=n_\mu n^\mu$. Torsion freedom gives $n^\mu\nabla_\mu n_\nu=\tfrac12\partial_\nu q$. Since $q$ vanishes on $\mathcal N$, all tangential derivatives vanish there and $dq=C\,du$ there for a scalar $C$. Consequently
$$
\ell^\mu\nabla_\mu\ell_\nu=\left(\ell(\log|f|)+\tfrac12fC\right)\ell_\nu.
$$
This is the unparametrized <null geodesic> equation. The integral curves are therefore generators of $\mathcal N$. An <affine rescaling of a null normal> removes the proportionality coefficient locally: if $\nabla_\ell\ell=\kappa\ell$, set $K=h\ell$ with $\ell(\log h)=-\kappa$.
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