Backlund transformation of Painleve III($D_8$) tau function

Understand

We study explicit formula (suggested by Gamayun, Iorgov, Lisovyy) for Painlev\'e III($D_8$) $\tau$ function in terms of Virasoro conformal blocks with central charge $1$.

• The Painlev\'e equation has two types of bilinear forms, we call them Toda-like and Okamoto-like.
• We obtain these equations from the representation theory using an embedding of a direct sum of two Virasoro algebra in a certain superalgebra.
• These two types of bilinear forms correspond to Neveu-Schwarz sector and Ramond sector of this algebra.