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

Negative flows of generalized KdV and mKdV hierarchies and their gauge-Miura transformations