A kind of nonisospectral and isospectral integrable couplings and their Hamiltonian systems

We first introduce a Lie algebra g ($) over tilde which can be used to construct integrable couplings of some isospectral and nonisospectral problems. As two applications of the Lie algebra g ($) over tilde, the MKdV spectral problem is enlarged to an isospectral problem and the AKNS spectral problem is expanded to a nonisopectral problem. Then, two integrable couplings are obtained by solving an isospectral and a nonisospectral zero-curvature equations. We find that the two hierarchies that we obtain have bi-Hamiltonian structure of combinatorial form. Additionally, some symmetries and conserved quantities of the resulting hierarchy are investigated. (c) 2021 Elsevier B.V. All rights reserved.