On the Uniqueness of the Schwarzian and Linearisation by Nonlocal Contact Transformation

We investigate the uniqueness of the Schwarzian from a group theoretic point of view. The Kummer]Schwarz equation, obtained from the Schwarzian, is also analysed for its group properties. Related equations arising in the study of geodesic curves on a surface of constant curvature. are constructed. These equations are generalised by reducing the symmetry imposed. The unique generalisation is linearised under a nonlocal contact transformation