  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
$A^{(1)}_{N-1}$ invariant
  Painlev\'e equations. The $A^{(1)}_{N-1}$ B\"acklund symmetries of dressing
equations and Painlev\'e equations are obtained in the self-similarity limit of
gauge transformations of the mKdV hierarchy realized as zero-curvature
equations on the loop algebra $\widehat{sl}(N)$ endowed with a principal
gradation.
