Infinite-dimensional And Finite-dimensional Integrable And Superintegrable Systems

Integrable and superintegrable spectral problems are constructed based on loop algebra; we get the associated Lax pairs and by the compatible condition of the Lax pairs, we derive the nonlinear evolution equation hierachy:The Hamiltonian equations are given with the help of the trace identity; With the help of classical mechanics, a reasonable Jacobi-Ostrogradsky coordinate system is found. and the finite-dimentional Hamiltonian equations are obtained; We construct a real Hamiltonian system which is equavallent to the complex Hamiltonian system:By the confocal system and the linear independent involution systems, the completely integrability of the finite system is proved.