What does integrability of finite-gap or 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-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 also discussed.