A new Lie algebra along with its induced Lie algebra and associated with applications

A 3 × 3 Lie algebra H is introduced whose induced Lie algebra by decomposition and linear combinations is obtained, which may reduce to the Lie algebra given by AP Fordy and J Gibbons. By employing the induced Lie algebra and the zero curvature equation, a kind of enlarged Boussinesq soliton hierarchy is produced. Again making use of a subalgebra of the induced Lie algebra leads to the well-known KdV hierarchy whose expanding integrable system is also worked out. As an applied example of the Lie algebra H, we obtain a new integrable coupling of the well-known AKNS hierarchy.