  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.
