Rigid equivalences of $5$-dimensional $2$-nondegenerate rigid real hypersurfaces $M^5\subset {\mathbb{C}}^3$ of constant Levi rank $1$

We study the local equivalence problem for real-analytic (${\mathcal{C}}^{\omega }$) hypersurfaces $M^5\subset {\mathbb{C}}^3$ which, in coordinates $(z_1,z_2,w)\in {\mathbb{C}}^3$ with $w=u+iv$, are rigid: $$u=F(z_1,z_2,{\bar{z}}_1,{\bar{z}}_2),$$ with $F$ independent of $v$. Specifically, we study the group ${\mathsf{H}\mathsf{o}\mathsf{l}}_{\mathsf{r}\mathsf{i}\mathsf{g}\mathsf{i}\mathsf{d}}(M)$ of rigid local biholomorphic transformations of the form: $$(z_1,z_2,w)⟼(f_1(z_1,z_2),f_2(z_1,z_2),aw+g(z_1,z_2)),$$ where $a\in \mathbb{R}\mathrm{\backslash }\{0\}$ and $\frac{D(f_1,f_2)}{D(z_1,z_2)}\mathrm{\ne }0$, which preserve rigidity of hypersurfaces. After performing a Cartan-type reduction to an appropriate $\{e\}$-structure, we find exactly two primary invariants $I_0$ and $V_0$, which we express explicitly in terms of the $5$-jet of the graphing function $F$ of $M$. The identical vanishing $0\equiv I_0\left(J^{5}F\right)\equiv V_0\left(J^{5}F\right)$ then provides a necessary and sufficient condition for $M$ to be locally rigidly-biholomorphic to the known model hypersurface: $$M_{\mathsf{L}\mathsf{C}}\text{\hspace{0.17em}⁣}:u=\frac{z_1{\bar{z}}_1+\frac{1}{2}{z}_1^2{\bar{z}}_2+\frac{1}{2}{\bar{z}}_1^2{z}_2}{1-z_2{\bar{z}}_2}\mathrm{.}$$ We establish that $\dim, {\sf Hol}{\sf rigid} (M) \leq 7 = \dim, {\sf Hol}{\sf rigid} \big( M_{\sf LC} \big)$ always. If one of these two primary invariants $I_0\equiv ̸0$ or $V_0 \not\equiv 0$ does not vanish identically, we show that this rigid equivalence problem between rigid hypersurfaces reduces to an equivalence problem for a certain $5$-dimensional $\{e\}$-structure on $M$.