On the properties of Bäcklund transformations on Painleväé 6th equations

Abstract

In his series of papers Okamoto give detailed explanation of Painlevé equations of types 2-6. On the other hand Kajiwara et.al described Bäcklund transformations for these types of Painlevé equations. Later, Bobenko and Eitner found that Painlevé equations of types 6 and 5 give rise Bonnet surfaces and vise-versa. In this research we study properties of Bäcklund transformations of 6th Painlevé equations P6 and determine explicit formula of such transformations which give a new Bonnet surfaces.