  We continue the study of the Korteweg-de Vries equation in terms of
cento-affine curves, initiated by U. Pinkall. A centro-affine curve is a closed
parametric curve in the affine plane such that the determinant made by the
position and the velocity vectors is identically 1. The space of centro-affine
curves is acted upon by the special linear group, and the quotient is
identified with the space of Hill's equations with periodic solutions. It is
known that the space of centro-affine curves carries two pre-symplectic
structures, and the KdV flow is identified with is a bi-Hamiltonian dynamical
system therein. We introduce a 1-parameter family of transformations on
centro-affine curves, prove that they preserve both presymplectic structures,
commute with the KdV flow, and share the integrals with it. Furthermore, the
transformation commute with each other (Bianchi permutability). We also
describe integrals of the KdV equation as arising from the monodromy of Riccati
equations associated with centro-affine curves. We are motivated by our work in
progress (joint with M. Arnold, D. Fuchs, and I. Izmenstiev), concerning the
cross-ratio dynamics on ideal polygons in the hyperbolic plane and hyperbolic
space, whose continuous version is studied in the present note.
