On Mobius-invariant and symmetry-integrable evolution equations and the Schwarzian derivative

We consider symmetry-integrable evolution equations in 1 + 1 dimensions of order 3 and order 5. We show that there exist only three equations in this class that are invariant under the Mobius transformation, and we name those Schwarzian equations. We report an interesting relation between the recursion operators of the Schwarzian equations and the corresponding adjoint operators that generate hierarchies of Schwarzian systems in terms of the Schwarzian derivative. This indicates a deep relation between the Schwarzian equations and the Schwarzian derivative. A classification of the fully nonlinear third-order Schwarzian equations is also reported.