Symmetries in the third Painlev\'e equation arising from the modified Pohlmeyer-Lund-Regge hierarchy

Abstract

We propose a modification of the AKNS hierarchy that includes the "modified" Pohlmeyer-Lund-Regge (mPLR) equation. Similarity reductions of this hierarchy give the second, third, and fourth Painlev\'e equations. Especially, we present a new Lax representation and a complete description of the symmetry of the third Painlev\'e equation through the similarity reduction. We also show the relation between the tau-function of the mPLR hierarchy and Painlev\'e equations.