  For a polygon $x=(x_j)_{j\in \mathbb{Z}}$ in $\mathbb{R}^n$ we consider the
midpoints polygon $(M(x))_j=\left(x_j+x_{j+1}\right)/2\,.$ We call a polygon a
soliton of the midpoints mapping $M$ if its midpoints polygon is the image of
the polygon under an invertible affine map. We show that a large class of these
polygons lie on an orbit of a one-parameter subgroup of the affine group acting
on $\mathbb{R}^n.$ These smooth curves are also characterized as solutions of
the differential equation $\dot{c}(t)=Bc (t)+d$ for a matrix $B$ and a vector
$d.$ For $n=2$ these curves are curves of constant generalized-affine curvature
$k_{ga}=k_{ga}(B)$ depending on $B$ parametrized by generalized-affine arc
length unless they are parametrizations of a parabola, an ellipse, or a
hyperbola.
