A new 4-dimensional implicit vector-form loop algebra with arbitrary constants and the corresponding computing formula of constant γ in the Variation identity

A new 4-dimensional implicit vector-form loop algebra containing several arbitrary constants is presented, and the corresponding computing formula of constant γ in the Variation identity is obtained in this paper. As an application, a new Liouville integrable hierarchy is gotten which can be reduced to Dirac hierarchy, nonlinear Scho¨inger equation (NLS), M-KdV equation and so on.