Integrable coupling hierarchy and Hamiltonian structure for a matrix spectral problem with arbitrary-order

We presented an integrable coupling hierarchy of a matrix spectral problem with arbitrary order zero matrix r by using semi-direct sums of matrix Lie algebra. The Hamiltonian structure of the resulting integrable couplings hierarchy is established by means of the component trace identities. As an example, when r is 2 × 2 zero matrix specially, the integrable coupling hierarchy and its Hamiltonian structure of the matrix spectral problem are computed.

► An integrable coupling hierarchy of matrix spectral problem with an arbitrary order zero matrix r was constructed.
► The Hamiltonian structure of the integrable coupling hierarchy was presented.
► The results enriched multi-component integrable equations.