EQUATION OF MATHEMATICAL PHYSICS AND OTHER AREAS OF APPLICATION: A CLASSICAL INTEGRABLE SYSTEM AND THE INVOLUTIVE REPRESENTATION OF SOLUTIONS OF THE HIGHER ORDER KAUP-NEWELL EQUATION

Under the constrained condition induced by the eigenfunction expresson of the potential (u, v)~T=(-<A~2q, q>, <A~2p, p>)~T=f(q, p), the spatial part of the Lax pair of the Kaup-Newell equation is non linearized to be a completely integrable system (R2N, Adp∧dq, H=H1) with the HamiltonianH1+-<A~q,p>-(1/2)<A~2p,p><A~2q,q>.while the nonlinearization of the time part leads to its N-involutive system {Hm}. The involutive solution of the compatible fsystem (H1), (Hm) is mapped by into the solution of the higher order Kaup-Newell equation.