Polygonal bicycle paths and the Darboux transformation

Abstract

A bicycle $\left(n,k\right)$-gon is an equilateral $n$-gon whose $k$ diagonals are of equal length. In this paper we introduce periodic bicycle $\left(n,k\right)$-paths, which are a natural variation in which the polygon is replaced with a periodic polygonal path, and study their rigidity and integrals of motion.