International Journal of Theoretical Physics, Vol. 28, No. 8, 1989 Kac-Moody Algebra, Nonlocal Symmetries, and Backlund Transformation for KdV Equation A. Roy Chowdhury I and Swapna Roy ~ Received February 2, 1989 Gauge transformation of the Lax eigenfunction through the explicit use of Lie group generators is seen to generate a two-parameter Backlund transformation. Explicit integration of this in two particular cases leads to sech2 0 type and rational solutions starting from the trivial one. A method is indicated to generate infinitesimal transformations around u in the sense of Steudel, which in this case leads to a nonlocal structure of transformations. Lie algebra has been successfully used in the study of nonlinear integrable systems over the last two decades (Olshanetsky and Perelomov, 1981; Kupershmidt, 1987). Later, affine Kac-Moody algebras were also incorporated and elegant results were obtained by various authors (Kupershmidt and Wilson, 1985; Guil, 1984, 1985; Szmiglielski, 1988). Among the most important results are deformation, the Miura map, and the Backlund transformation, deduced via the automorphism of the Lie algebra. There has also been considerable study on the gauge transformation theory for nonlinear systems (Eichenherr and Honnerkamp, 1981; Hon- nerkamp, 1981). Here we show that it is possible to deduce the Backlund transformation for the KdV problem with an explicit realization of the gauge transformation of the Lax eigenfunction in a particular Lie algebra. To start with, let us consider the prolongation structure associated with the KdV equation (Van Eck, 1983): u, + 12uux + uxxx = 0 (1) written as w=dy+ A dx + Bdt J High Energy Physics Division, Department of Physics, Jadavpur University, Calcutta-700032, India. 845 0020-7748/89/0800-0845506.00/0 9 1989 Plenum Publishing Corporation