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

Gauge Miura and Backlund Transformations for Generalized $A_n$-KdV Hierarchies