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Explicit absolute parallelism for $2$-nondegenerate real hypersurfaces $M^5 \subset \mathbb{C}^3$ of constant Levi rank $1$

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 $M$.