1 M ay 2 00 5 What does integrability of finite - gap / soliton potentials mean ?

In the example of the Schrödinger/KdV equation we treat the theory as equivalence of two concepts of Liouvillian integrability: quadrature integrability of linear differential equations with a parameter (spectral problem) and Liouville’s integrability of finite-dimensional Hamiltonian systems (stationary KdV–equations). Three key objects in this field: the explicit Ψ-function, trace formula and the Jacobi problem provide a complete solution. The Θ-function language is derivable from these objects and used for ultimate representation of a solution to the inversion problem. Relations (transformations) with nonintegrable equations are discussed also.