Bäcklund transformation of Painlevé III( D ₈) τ function

We study the explicit formula (suggested by Gamayun, Iorgov and Lisovyy) for the Painlevé III( D ₈) τ function in terms of Virasoro conformal blocks with a central charge of 1. The Painlevé equation has two types of bilinear forms, which we call Toda-like and Okamoto-like. We obtain these equations from the representation theory using an embedding of the direct sum of two Virasoro algebras in a certain superalgebra. These two types of bilinear forms correspond to the Neveu–Schwarz sector and the Ramond sector of this algebra. We also obtain the τ functions of the algebraic solutions of the Painlevé III( D ₈) from the special representations of the Virasoro algebra of the highest weight ( n  +  1/4)². (paper)