Two types of loop algebras and their expanding Lax integrable models

Though various integrablehierarchies of evolution equations were obtained by choosingproper U in zero-curvature equation U_t-V_x+[U,V]=0, but in this paper, a new integrable hierarchy possessing bi-Hamiltonian structure is worked outby selecting V with spectral potentials. Then its expanding Lax integrable model of the hierarchy possessing a simple Hamiltonian operator widetilde{J} is presentedby constructing a subalgebrawidetilde{G } of the loop algebra widetilde A₂. Aslinear expansions of the above-mentioned integrable hierarchy andits expanding Lax integrable model with respect to theirdimensional numbers, their (2+1)-dimensional forms are derivedfrom a (2+1)-dimensional zero-curvature equation.