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The class ${\sf IV}_2$ of $2$-nondegenerate constant Levi rank $1$hypersurfaces $M^5 \subset \mathbb{C}^3$ is governed by Pocchiola's two primaryinvariants $W_0$ and $J_0$. Their vanishing characterizes equivalence of such ahypersurface $M^5$ to the tube $M_{\sf LC}^5$ over the real light cone in$\mathbb{R}^3$. When either $W_0 \not\equiv 0$ or $J_0 \not\equiv 0$, bynormalization of certain two group parameters ${\sf c}$ and ${\sf e}$, aninvariant coframe can be built on $M^5$, showing that the dimension of the CRautomorphism group drops from $10$ to $5$.  This paper constructs an explicit $\{e\}$-structure in case $W_0$ and $J_0$do not necessarily vanish. Furthermore, Pocchiola's calculations hidden on acomputer now appear in details, especially the determination of a secondaryinvariant $R$, expressed in terms of the first jet of $W_0$. All othersecondary invariants of the $\{e\}$-structure are also expressed explicitly interms of $W_0$ and $J_0$.

Differential $\{e\}$-structures for equivalences of $2$-nondegenerate Levi rank $1$ hypersurfaces $M^5 \subset \mathbb{C}^3$