Kimura, Hironobu; Okamoto, Kazuo On the polynomial Hamiltonian structure of the Garnier systems. (English) Zbl 0562.34004 J. Math. Pures Appl., IX. Sér. 63, 129-146 (1984). 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 \(s_ 1,s_ 2,...,s_ N\). Reviewer: B.Rodriguez-Salinas Cited in 13 Documents MSC: 34A30 Linear ordinary differential equations and systems Keywords:Painlevé equation; isomonodromic deformation equation; Garnier system PDFBibTeX XMLCite \textit{H. Kimura} and \textit{K. Okamoto}, J. Math. Pures Appl. (9) 63, 129--146 (1984; Zbl 0562.34004)