A new Kaup-Newell type soliton hierarchy is generated from an asymmetric matrix spectral problem associated with the three-dimensional special linear Lie algebra sl(2,R). Then based on semi-direct sums of matrix Lie algebras consisting of 3. ×. 3 block matrix Lie algebras, corresponding bi-integrable couplings of this hierarchy are constructed. Each equation in the resulting system has a bi-Hamiltonian structure furnished by the variational identity, which lead to Liouville integrability.
Yu, S., Yao, Y., Shen, S., & Ma, W. X. (2015). Bi-integrable couplings of a Kaup-Newell type soliton hierarchy and their bi-Hamiltonian structures. Communications in Nonlinear Science and Numerical Simulation, 23(1–3), 366–377. https://doi.org/10.1016/j.cnsns.2014.12.008