  We study the local equivalence problem for five dimensional real
hypersurfaces $M^5$ of $\mathbb{C}^3$ which are $2$-nondegenerate and of
constant Levi rank $1$ under biholomorphisms. We find two invariants, $J$ and
$W$, which are expressed explicitly in terms of the graphing function $F$ of
$M$, the annulation of which give a necessary and sufficient condition for $M$
to be locally biholomorphic to a model hypersurface, the tube over the light
cone. If one of the two invariants $J$ or $W$ does not vanish on $M$, we show
that the equivalence problem under biholomophisms reduces to an equivalence
problem between $\{e \}$-structures, that is we construct an absolute
parallelism on $M$.
