An integrable hierarchy and Darboux transformations, bilinear Bäcklund transformations of a reduced equation

A new integrable hierarchy of evolution equations is obtained by making use of a Lie algebra and Tu–Ma scheme, from which a new generalized Broer–Kaup (gBK) equation is produced. Then two kinds of Darboux transformations, the bilinear presentation, the bilinear Bäcklund transformation and the new Lax pair of the gBK equation are generated, respectively, by employing the Bell polynomials.