Hierarchies, integrable decompositions and conservation laws of Geng equation

In this paper, two hierarchies of the Geng equations are presented, including positive non-isospectral hierarchy and negative non-isospectral hierarchy. Moreover, integrable couplings of the corresponding Geng hierarchies are also constructed by enlarging the associated matrix spectral problem. Three new integrable decompositions and conservation laws of the isospectral Geng equation are also obtained. The Gauge transformations are used to obtain the associated binary symmetry constraints of the Geng equation at the first time.