A type of new integrable Hamiltonian hierarchy and expanding integrable model of its reduced integrable system associated with a new loop algebra

Designing a new isospectral problem from a loop algebra . It follows that a type of new integrable hierarchy of evolution equations, possessing bi-Hamiltonian structure, is obtained. As a reduction case, the well-known TC hierarchy is presented. Then a new seven-dimensional loop algebra is constructed. From which, expanding integrable model of the TC hierarchy is obtained.