Gauge Miura and Bäcklund transformations for generalized A _n-KdV hierarchies

The construction of Miura and B\"acklund transformations for $A_n$ mKdV KdV hierarchies are presented in terms gauge acting upon the zero curvature representation. As well known $sl(2)$ case, we derive relate equations motion two hierarchies. Moreover, Miura-gauge transformation is not unique, instead, it shown to be connected a set generators labeled by exponents generalized gauge-B\"acklund $A_n$-KdV hierarchy obtained as composition B\"acklund-gauge $A_n$-mKdV hierarchy. representation provide framework which universal within all flows generate systematically entirely