Geometric integrability of the Camassa-Holm equation

It is observed that the Camassa-Holm equation describes pseudo-spherical surfaces and that therefore, its integrability properties can be studied by geometrical means. An sl(2, R)-valued linear problem whose integrability condition is the Camassa-Holm equation is presented, a 'Miura transform' and a 'modified Camassa-Holm equation' are introduced, and conservation laws for the Camassa-Holm equation are then directly constructed. Finally, it is pointed out that this equation possesses a nonlocal symmetry, and its flow is explicitly computed.