Diff(S-1) and Adler-Gelfand-Dikii spaces and integrable systems

We study a family of Korteweg-deVries equations as some evolution equations associated to the Adler-Gelfand-Dikii (AGD) space. First we derive (formal ly) the Korteweg-de Vries (KdV) as an evolution equation of the AGD operato r under the action of Vect(S-1). The solutions of the AGD operator define a n immersion C --> CPn-1 in homogeneous coordinates. We derive the Schwarzia n KdV equation as an evolution of the solution curve associated to Delta ((n))=d(n)/ dx(n)+u(n-2) d(n-2)/dx(n-2) +...+u(0). This equation is invariant under linear fractional transformations. We also show how the modified KdV is related to the Schwarzian KdV by the Cole-Hop f transformation. The geometrical (differential Galois theory) connections between all these equations are given.