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 $w = u+i, v$, are rigid: [ u ,=, Fbig(z_1,z_2,overline{z}_1,overline{z}_2big), ] with $F$ independent of $v$. Specifically, we study the group ${sf Hol}_{sf rigid}(M)$ of rigid local biholomorphic transformations of the form: [ big(z_1,z_2,wbig) longmapsto Big( f_1(z_1,z_2), f_2(z_1,z_2), a,w + g(z_1,z_2) Big), ] where $a in mathbb{R} backslash {0}$ and $frac{D(f_1,f_2)}{D(z_1,z_2)} eq 0$, 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 $0 equiv I_0 big( J^5F big) equiv V_0 big( J^5F big)$ then provides a necessary and sufficient condition for $M$ to be locally rigidly-biholomorphic to the known model hypersurface: [ M_{sf LC} colon u ,=, frac{z_1,overline{z}_1 +frac{1}{2},z_1^2overline{z}_2 +frac{1}{2},overline{z}_1^2z_2}{ 1-z_2overline{z}_2}. ] 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 $I_0 otequiv 0$ or $V_0 otequiv 0$ 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 $M$.
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