Complex form, reduction and Lie–Poisson structure for the nonlinearized spectral problem of the Heisenberg hierarchy

Abstract

In this paper, the relation between the different restricted systems associated with the Heisenberg magnetic (HM) equation is studied by the reduction procedure. With the help of a Lie group homomorphism of SU (2) into SO (3) , the Euler–Rodriguez-type parameters are introduced to generate new finite-dimensional integrable system. It has shown that the resulting system, which is the nonlinearized spectral problem of HM hierarchy on C 2 N , is a Hamiltonian system in complex form. Further, Poisson reduction and Lie–Poisson structure are derived by the method of invariants. The reduced system is found to be a Hamiltonian system on the orbit space C 2 N / T N ≃ R 3 N , coinciding with the nonlinearized Lenard spectral problem. Moreover, the fully reduced systems on the leaves of the symplectic foliation are also given. Specifically, the reduction extended to the common level set of the complex cones is the usual 2×2 nonlinearized spectral problem. Finally, the integrability of the system with Lie–Poisson structure is proven by making use of the SO (3) symmetry.