What does integrability of finite-gap or soliton potentials mean?

Abstract

In the example of the Schr\"odinger/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: new explicit $\Psi$-function, trace formula and the Jacobi problem provide a complete solution. The $\Theta$-function language is derivable from these objects and used for ultimate representation of a solution to the inversion problem. Relations with non-integrable equations are discussed also.