Diff (S¹) 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 (formally) the Korteweg-deVries (KdV) as an evolution equation of the AGD operator under the action of Vect(S¹).The solutions of the AGD operator define an immersion C → ℂP^n-1 in homogeneous coordinates. We derive the Schwarzian KdV equation as an evolution of the solution curve associated to Δ⁽ⁿ⁾= dⁿ/dxⁿ+u_n-2 d^n-2/dx^n-2 + ... + u₀. This equation is invariant under linear fractional transformations. We also show how the modified KdV is related to the Schwarzian KdV by the Cole-Hopf transformation. The geometrical (differential Galois theory) connections between all these equations are given.