Korteweg-de Vries hierarchy and related completely integrable systems: I. Algebro-geometrical approach

Abstract

We consider complementary dynamical systems related to stationary Korteweg-de Vries hierarchy of equations. A general approach for finding elliptic solutions is given. The solutions are expressed in terms of Novikov polynomials in general quais-periodic case. For periodic case these polynomials coincide with Hermite and Lam\'e polynomials. As byproduct we derive $2\times 2$ matrix Lax representation for Rosochatius-Wojciechiwski, Rosochatius, second flow of stationary nonlinear vectro Schr\"{o}dinger equations and complex Neumann system.