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In a previous memoir 2202.03030, we showed that in every dimension $n \geq5$, there exists (unexpectedly) no affinely homogeneous hypersurface $H^n\subset \mathbb{R}^{n+1}$ having Hessian of constant rank 1 (and not beingaffinely equivalent to a product with $\mathbb{R}^{m \geqslant 1}$). Thepresent work is devoted to determine all non-product constant Hessian rank 1affinely homogeneous hypersurfaces $H^n \subset \mathbb{R}^{n+1}$ in dimensions$n = 2, 3, 4$, the cases $n = 1, 2$ being known. With complete details in thecase $n = 2$, we illustrate the main features of what can be termed the "PowerSeries Method of Equivalence". The gist is to capture invariants at the originonly, to create branches, and to infinitesimalize calculations. In dimension $n= 3$, we find a single homogeneous model: \[ u \,=\, \frac{1}{3\,z^2} \Big\{\big( 1-2\,y+y^2-2\,xz \big)^{3/2} - (1-y)\, \big( 1-2\,y+y^2-3\,xz \big)\Big\}, \] the singularity $\frac{1}{3z^2}$ being illusory. In dimension $n =4$, without reaching closed forms, we find two simply homogeneous models,differing by some $\pm$ sign.

Classification of Hessian Rank 1 Affinely Homogeneous Hypersurfaces H^n in R^{n+1} in Dimensions n = 2, 3, 4