Integrable motions of space curves in affine geometry

It is shown that many well-known integrable equations such as the KdV, Harry Dym, Sawada–Kotera hierarchies and the Kaup–Kupershmidt, Boussinesq, Tzitzéica, Hirota–Satsuma equations naturally arise from motions of space curves in affine and centro-affine geometries. The motions of curves corresponding to travelling waves of the KdV and Sawada–Kotera equations are constructed explicitly.