For each Painlevé system 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 are successively obtained from that for by the well known degeneration or confluence processes.
Masaki SUZUKI. Nobuhiko TAHARA. Kyoichi TAKANO. "Hierarchy of Bäcklund transformation groups of the Painlevé systems." J. Math. Soc. Japan 56 (4) 1221 - 1232, October, 2004. https://doi.org/10.2969/jmsj/1190905457