TOWARDS AN ALGEBRAIC APPROACH TO INTEGRABLE SYSTEMS

§ l.A quarter of century ago the discovery of solitary waves ( soliton) solutions to Korteweg-de Vries equations opened a path for a reviewing of the phenomena associated with integrability. Nowadays, several aspects of integrability of finite and infinite dimen sional systems have grown apart and relate to Lie group Theory, Algebraic Geometry, infinite dimensional Lie algebras, infinite dimensional grasmannians and tau functions. In classical hamiltonian mechanics, the complete integrability over the reals is explained by Arnold [Ar]. If we start with a system of Ordinary Differential Equations: i = J(z).gradH, where J ( z) = skew-symmetric matrix with polynomial entries in Z; and the Poisson bracket: {f, g}: = < gradf, J(z).gradg> satisfies the Jacobi identity, then there exist (k+m) independent integrals HI,..., H k+m in involution ( i.e. {Hi, Hj} = 0). The first k invariants are called trivial or Casimir invariants since J(z).gradHi = O. The remaining integrals give rise to m commuting vector fields: Xi: it = J (z) .gradHi+k constrained to the variety n7�t {Hi = c; }. According to th