Hierarchy of Bäcklund transformation groups of the Painlevé systems

Abstract

For each Painlevé system $P_J$ except the first one, we have a Bäcklund transformation group which is a lift of an alfine Weyl group. In this PaPer, we show that the Bäcklund transformation groups for $J=V,$$IV,$$III,$$II$ are successively obtained from that for $J=VI$ by the well known degeneration or confluence processes.