Twenty five years ago U. Pinkall discovered that the Korteweg-de Vries equation can be realized as an evolution of curves in centoraffine geometry. Since then, a number of authors interpreted various properties of KdV and its generalizations in terms of centoraffine geometry. In particular, the Backlund transformation of the Korteweg-de Vries equation can be viewed as a relation between centroaffine curves. Our paper concerns self-Backlund centroaffine curves. We describe general properties of these curves and provide a detailed description of them in terms of elliptic functions. Our work is a centroaffine counterpart to the study done by F. Wegner of a similar problem in Euclidean geometry, related to Ulams problem of describing the (2-dimensional) bodies that float in equilibrium in all positions and to bicycle kinematics. We also consider a discretization of the problem where curves are replaced by polygons. This is related to discretization of KdV and the cross-ratio dynamics on ideal polygons.
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