Letters in Mathematical Physics 9 (1985) 85-91. 0377-9017j85.15. @ 1985 by D. Reidel Publishing Company. 85 A Geometrical Approach to the Integrability of Soliton Equations S. DE FILIPPO*, M. SALERNO* Laboratory of Applied Mathematical Physics, The Technical University of Denmark, DK-2800 Lyngby, Denmark and G. VILASI Istituto di Fisica, Universitgt di Salerno, 1-84100 Salerno, Italy (Received: 1 December, 1983) Abstract. A separability criterion in one-degree-of-freedom dynamics, suitable for soliton equations, is given in terms of a geometrical structure on the phase manifold. For solitonic degrees of freedom, i.e., those corresponding to the discrete spectrum of the associated Lax operator, integrability is a priori proved. 1. The inverse scattering transform method (IST) based on Lax representation is universally recognized as one of the standard techniques in integrating partial differential equations [ 1]. In spite of its success as an integration algorithm, a compact a priori criterion of integrability in terms of Lax pairs is, to date, lacking. On the other hand, IST being a transformation from generic coordinates (potentials) to action-angle variables [2], makes it only natural for us to seek an integrability criterion for soliton equations by looking at them as dynamical systems on (infinite-dimensional) phase manifolds [3-6]. This point of view is also suggested by the occurrence in the IST of a peculiar operator (the squared eigenfunctions operator) [7], relevant for the effec- tiveness of the method, which naturally fits in this geometrical setting as a mixed tensor field on the phase manifold M. Such a tensor operator, named in the literature in several ways (squared eigen- functions operator, hereditary operator, recursion operator .... ), satisfies the following general properties [3-9]: (i) it is invariant under the dynamics (this is also known as T being a strong symmetry), (ii) its Nijhenhuis tensor vanishes (also known as hereditary property), (iii) it has a doubly degenerate continuous spectrum and finitely many discrete eigenvalues (whose number fixes the soliton sector, which means a fuxed connected component of M) with bidimensional invariant spaces. Each one of them is spanned by a true eigenvector and a generalized one; this corresponding "* Permanent address. Istituto di Fisica, Universit~t di Salerno, 1-84100 Salerno, Italy.