Letters in Mathematical Physics 19: 35-44, 1990. 1990 Kluwer Academic Publishers. Printed in the Netherlands. 35 Loop Algebras and their Relation to the Conformal Structure of Integrable Systems SVERRIR OLAFSSON* Department of Mathematics, The University of Manchester, Institute of Science and Technology, P.O. Box 88, Manchester M60 IQD, U.K. (Received: 17 April 1989; revised version: 3 July 1989) A~traet. We use the theorem of Kostant, Adler and Symes to construct an infinite set of local polynomials in involution with respect to the Poisson bracket realisation of the Neveu-Schwartz sector of the N = I superconformal algebra. AMS subject classifications (1980). 17B67, 58F07, 81E13. The Virasoro algebra and its supersymmetric extension, the superconformal alge- bra, are of basic importance in several branches of mathematical physics. In string theories, they appear as the residual symmetry of the reparametrization invariance of the world sheet. Further, the conformal algebra turns up in the theory of critical phenomena as a result of the scaling invariance in the critical point. Gervais [ 1] and Kupershmidt [2] have shown how the Virasoro algebra appears in connection with the KdV equation which suggests its relation to the theory of integrable systems. Kupershmidt [3], Chaichian and Kulish [4], Mathieu [5], Ho-Kim [6], Bilal and Gervais [7] and other authors, have worked on the extension of these ideas to include superintegrable equations. Recently, Olafsson [8] used the theorem of Kostant, Adler and Symes [9] to construct super nonlinear evolution equations associated with loop algebras = Ej~z g<J), g(J) = g | 2J, g finite-dimensional simple super Lie algebra. Some of the equations found are identical to equations known in the literature [10] and references therein, others are new. In [10], the Hamiltonian structure of the equations was not discussed nor their conservation laws. This was done in [8] and the conserved quantities were shown to be local polynomials in k-- dim(g)- rank(g) fields and their x-derivatives. These Poisson brackets commute with respect to a given canonical structure. In this Letter, we show how different restrictions on the fields lead to infinite local polynomials in involution with respect to new canonical structures. In one case, this structure is given by a Poisson bracket realisation of the Neveu-Schwartz (NS) * Present address: Faculty of Mathematical Studies, University of Southampton, Southampton S09 5NH, U.K.