On the transformation group of the second Painlevé equation

Abstract

We show that for the second Painlevé equation $y'' = 2y^{3}+ty+\alpha$, the Bäcklund transformation group $G$, which is isomorphic to the extended affine Weyl group of type $\hat{A}_{1}$, operates regularly on the natural projectification ${\mathcal X}(c)/\mathbb{C}(c, t)$ of the space of initial conditions, where $c = \alpha-1/2$. ${\cal X}(c)/\mathbb{C}(c, t)$ has a natural model ${\mathcal X}[c]/\mathbb{C}(t)[c]$. The group $G$ does not operate, however, regularly on $\mathcal {X}[c]/\mathbb{C}(t)[c]$. To have a family of projective surfaces over $\mathbb{C}(t)[c]$ on which $G$ operates regularly, we have to blow up the model ${\mathcal X}[c]$ along the projective lines corresponding to the Riccati type solutions.