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Studies on the Painlevé equations. IV: Third Painlevé equation $P\sb{III}$. (English) Zbl 0639.58013
Summary: [For part III see Math. Ann. 275, 221-255 (1986; Zbl 0589.58008).] The present article deals with the third Painlevé equation $P\sb{III}$; we consider instead the equation $P\sb{III'}$, equivalent to the former. By defining the Painlevé system ${\cal H}$, we consider the group $G\sb*$ of birational canonical transformations of ${\cal H}$; $G\sb*$ is isomorphic to the affine Weyl group of the root system of the type $B\sb 2$. A sequence of solutions of ${\cal H}$ is obtained from that of $\tau$-functions, satisfying the Toda equation and vice versa. We consider also particular solutions of ${\cal H}$ written in terms of the cylinder function.

37J99Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
14E05Rational and birational maps