  The construction of Integrable Hierarchies in terms of zero curvature
representation provides a systematic construction for a series of integrable
non-linear evolution equations (flows) which shares a common affine Lie
algebraic structure. The integrable hierarchies are then classified in terms of
a decomposition of the underlying affine Lie algebra $\hat {\cal{G}} $ into
graded subspaces defined by a grading operator $Q$. In this paper we shall
discuss explicitly the simplest case of the affine $\hat {sl}(2)$ Kac-Moody
algebra within the principal gradation given rise to the KdV and mKdV
hierarchies and extend to supersymmetric models.
  It is known that the positive mKdV sub-hierachy is associated to some
positive odd graded abelian subalgebra with elements denoted by $E^{(2n+1)}$.
Each of these elements in turn, defines a time evolution equation according to
time $t=t_{2n+1}$. An interesting observation is that for negative grades, the
zero curvature representation allows both, even or odd sub-hierarchies. In both
cases, the flows are non-local leading to integro-differential equations.
Whilst positive and negative odd sub-hierarchies admit zero vacuum solutions,
the negative even admits strictly non-zero vacuum solutions. Soliton solutions
can be constructed by gauge transforming the zero curvature from the vacuum
into a non trivial configuration (dressing method).
  Inspired by the dressing transformation method, we have constructed a
gauge-Miura transformation mapping mKdV into KdV flows. Interesting new results
concerns the negative grade sector of the mKdV hierarchy in which a double
degeneracy of flows (odd and its consecutive even) of mKdV are mapped into a
single odd KdV flow. These results are extended to supersymmetric hierarchies
based upon the affine $\hat {sl}(2,1)$ super-algebra.
