An integrable Hamiltonian hierarchy and its constrained flows with generalized Hamiltonian regular representations, as well as its expanding integrable system

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A new subalgebra of loop algebra A ̃ 2 is first constructed. It follows that an isospectral problem is established. Using Tu-pattern gives rise to a new integrable hierarchy, which possesses bi-Hamiltonian structure. As its reduction cases, the well-known standard Schrodinger equation and MKdV equation are presented, respectively. Furthermore, by making use of bi-symmetry constraints, generalized Hamiltonian regular representations for the hierarchy are obtained. At last, we obtain an expanding integrable system of this hierarchy by applying a scalar transformation between two isospectral problems and constructing a five-dimensional loop algebra G̃. In particular, the expanding integrable models of Schrodinger equation and MKdV equation are presented, respectively.