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 affine 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.