Gauge symmetry origin of Bäcklund transformations for Painlevé equations

We identify the self-similarity limit of the second flow of sl(N) mKdV hierarchy with the periodic dressing chain thus establishing a connection to $A_{N-1}^{\left(1\right)}$ invariant Painlevé equations. The $A_{N-1}^{\left(1\right)}$ Bäcklund symmetries of dressing equations and Painlevé equations are obtained in the self-similarity limit of gauge transformations of the mKdV hierarchy realized as zero-curvature equations on the loop algebra $\widehat{sl}\left(N\right)$ endowed with a principal gradation. (paper)