Historical remarks to finite-gap integration theory: elementary   treatment of the theory
  In the example of the Schr\"odinger/KdV equation we give elementary treatment of the theory of finite-gap integration. The concept is equivalent to two kinds 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 new explicit formula for the $\Psi$-function, trace formula and the Jacobi problem provide a complete solution.   Appendix contains Russian translation of two papers of J.Drach 
Yu.V.Brezhnev