  Equivalences under the affine group ${\rm Aff} (\mathbb{R}^3)$ of constant
Hessian rank $1$ surfaces $S^2 \subset \mathbb{R}^3$, sometimes called
parabolic, were, among other objects, studied by Doubrov, Komrakov, Rabinovich,
Eastwood, Ezhov, Olver, Chen, Merker, Arnaldsson, Valiquette. Especially,
homogeneous models and algebras of differential invariants in various branches
have been fully understood.
  Then what about higher dimensions? We consider hypersurfaces $H^n \subset
\mathbb{R}^{n+1}$ graphed as $\{ u = F(x_1, \dots, x_n) \}$ whose Hessian
matrix $(F_{x_i x_j})$, a relative affine invariant, is, similarly, of constant
rank $1$. Are there homogeneous models?
  Complete explorations were done by the author on a computer in dimensions $n
= 2, 3, 4, 5, 6, 7$. The first, expected outcome, was to obtain a complete
classification of homogeneous models in dimensions $n = 2, 3, 4$ (forthcoming
article, case $n = 2$ already known). The second, unexpected outcome, was that
in dimensions $n = 5, 6, 7$, there are no affinely homogenous models!
  (Except those that are affinely equivalent to a product of $\mathbb{R}^m$
with a homogeneous model in dimensions $2, 3, 4$.)
  The present article establishes such a non-existence result in every
dimension $n \geqslant 5$, based on the production of a normal form for $\{ u =
F(x_1, \dots, x_n) \}$ under ${\rm Aff} (\mathbb{R}^{n+1})$, up to order
$\leqslant n+5$, valid in any dimension $n \geqslant 2$.
