  The KdV hierarchy is a paradigmatic example of the rich mathematical
structure underlying integrable systems and has far-reaching connections in
several areas of theoretical physics. While the positive part of the KdV
hierarchy is well known, in this paper we consider an affine Lie algebraic
construction for its negative part. We show that the original Miura
transformation can be extended to a gauge transformation that implies several
new types of relations among the negative flows of the KdV and mKdV
hierarchies. Contrary to the positive flows, such a ``gauge-Miura''
correspondence becomes degenerate whereby more than one negative mKdV model is
mapped into a single negative KdV model. For instance, the sine-Gordon and
another negative mKdV flow are mapped into a single negative KdV flow which
inherits solutions of both former models. The gauge-Miura correspondence
implies a rich degeneracy regarding solutions of these hierarchies. We obtain
similar results for the generalized KdV and mKdV hierachies constructed with
the affine Lie algebra $\widehat{s\ell}(r+1)$. In this case the first negative
mKdV flow corresponds to an affine Toda field theory and the gauge-Miura
correspondence yields its KdV counterpart. In particular, we show explicitly a
KdV analog of the Tzitz\'{e}ica-Bullough-Dodd model. In short, we uncover a
rich mathematical structure for the negative flows of integrable hierarchies
obtaining novel relations and integrable systems.
