Three New Integrable Hierarchies of Equations

Abstract

A general Lie algebra Vs and the corresponding loop algebraṼs are constructed, from which the linear isospectralLax pairs are established, whose compatibility presents the zerocurvature equation. As its application, a new Lax integrablehierarchy containing two parameters is worked out. It is notLiouville-integrable, however, its two reduced systems areLiouville-integrable, whose Hamiltonian structures are derived bymaking use of the quadratic-form identity and the γformula (i.e. the computational formula on the constant γappeared in the trace identity and the quadratic-form identity).