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On the transformation group of the second Painlevé equation. (English) Zbl 0972.34073
For the second Painlevé equation $y''=2y^3+ty +\alpha, \quad \alpha\in\mathbb C,$ the Bäcklund transformation group is isomorphic to the extended affine Weyl group of type $$\widetilde{A}_1,$$ namely the group $$G$$ generated by the translations $$t_+(\alpha)=\alpha+1,$$ $$t_-(\alpha)=\alpha-1$$ and the reflection $$i(\alpha)=-\alpha.$$ For the affine spaces $$\mathbb A^4$$ and $$\mathbb A^2$$ with coordinate systems $$(y,y',t,\alpha)$$ and $$(t,\alpha),$$ respectively, consider the vector field $\delta(\alpha)=\frac{\partial}{\partial t}+y' \frac{\partial}{\partial y}+(2y^3+ty+\alpha) \frac{\partial}{\partial y'}$ on $$\mathbb A^4$$ and a natural fibration $$\pi : \mathbb A^4 \to \mathbb A^2$$ by $$(y,y',t,\alpha)\mapsto (t,\alpha).$$
Here, the author constructs a projective model of the fibration $$\pi : \mathbb A^4 \to \mathbb A^2$$ on which the group $$G$$ operates regularly.

##### MSC:
 34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
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##### References:
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