Integrable motions of curves in projective geometries

In this paper, integrable systems satisfied by the curvatures of curves under inextensible motions in projective geometries are identified. It is shown that motions of curves in two-, three- and four-dimensional projective geometries are respectively related to the Kaup–Kupershmidt hierarchy, a coupled Kaup–Kupershmidt–KdV system and their extension.