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A new Lax pair for the sixth Painlevé equation associated with \(\widehat{\mathfrak {so}} (8)\). (English) Zbl 1047.34105

Kawai, Takahiro (ed.) et al., Microlocal analysis and complex Fourier analysis. River Edge, NJ: World Scientific (ISBN 981-238-161-9/hbk). 238-252 (2002).
The authors present a new Lax pair for the classical sixth Painlevé equation \(P_{VI}\) which provides one with a natural isomonodromic explanation for all discrete symmetries of \(P_{VI}\).
A Bäcklund transformation group of \(P_{VI}\) consists of transformations of dependent variables and parameters that leaves an equivalent Hamiltonian system \(H_{VI}\) invariant. The so-called fundamental Bäcklund transformations \(s_j\), \(j=0,1,2,3,4\), generate subgroups isomorphic to the affine Weyl group of type \(D_4^{(1)}\) and, with transformations \(r_i\), \(i=1,3,4\), together, to the extended affine Weyl group. Using the classical Lax pair representation of rank two with 4 regular singular points associated with \(su(2)\), it is possible to explain transformations \(s_j\), \(j=0,1,3,4\), except for \(s_2\), as a result of certain gauge transformations of the Lax pair, while \(r_i\), \(i=1,3,4\), appear from fractional linear transformations of an auxiliary independent variable.
The authors find a Lax pair for the Hamiltonian system \(H_{VI}\) in \(8\times8\)-matrices with a regular singular point at the origin and a degenerate irregular singularity at infinity. This remarkable Lax pair allows one to obtain all fundamental Bäcklund transformations \(s_j\), \(j=0,1,2,3,4\), and \(r_i\), \(i=1,3,4\), using explicitly specified gauge transformations.
For the entire collection see [Zbl 1021.00004].

MSC:

34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
34M15 Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) of ordinary differential equations in the complex domain
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