An integrable Hamiltonian hierarchy, a high-dimensional loop algebra and associated integrable coupling system

A subalgebra of loop algebra à 2 is established. Therefore, a new isospectral problem is designed. By making use of Tu's scheme, a new integrable system is obtained, which possesses bi-Hamiltonian structure. As its reductions, a formalism similar to the well-known Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy and a generalized standard form of the Schrödinger equation are presented. In addition, in order for a kind of expanding integrable system to be obtained, a proper algebraic transformation is supplied to change loop algebra à 2 into loop algebra à 1 . Furthermore, a high-dimensional loop algebra is constructed, which is different from any previous one. An integrable coupling of the system obtained is given. Finally, the Hamiltonian form of a binary symmetric constrained flow of the system obtained is presented.