A new subalgebra of the Lie algebra A2 and two types of integrable Hamiltonian hierarchies, expanding integrable models

A new subalgebra G of the Lie algebra A2 is first constructed. Then two loop algebra , are presented in terms of different definitions of gradations. Using , designs two isospectral problems, respectively. Again utilizing Tu-pattern obtains two types of various integrable Hamiltonian hierarchies of evolution equations. As reduction cases, the well-known Schrödinger equation and MKdV equation are obtained. At last, we turn the subalgebras , of the loop algebra into equivalent subalgebras of the loop algebra by making a suitable linear transformation so that the two types of 5-dimensional loop algebras are constructed. Two kinds of integrable couplings of the obtained hierarchies are showed. Specially, the integrable couplings of Schrödinger equation and MKdV equation are obtained, respectively.