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On the polynomial Hamiltonian structure of the Garnier systems. (English) Zbl 0562.34004

It is well known that R. Fuchs derived the sixth Painlevé equation as an isomonodromic deformation equation of a linear differential equation of the second order. His result was extended by R. Garnier [Ann. Sc. Norm. Sup. Pisa, Cl. Sci. 29, 1-126 (1912)]. The purpose of this paper is to transform the Garnier system, according to Okamoto, by a canonical transformation into a system of the form \(\{dq_ j=\sum^{N}_{i=1}(\partial H_ i/\partial q_ j)ds_ i\), \(dp_ j=- \sum^{N}_{i=1}(\partial H_ i/\partial q_ j)ds_ i\) (1\(\leq j\leq N)\}\) such that the Hamiltonian functions \(H_ j\) (1\(\leq j\leq N)\) are polynomials in \(q_ 1,...,q_ N\), \(p_ 1,...,p_ N\) with coefficients rational in \(s_ 1,s_ 2,...,s_ N\).

MSC:

34A30 Linear ordinary differential equations and systems
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