Babich, M. V. On canonical parametrization of the phase spaces of equations of isomonodromic deformations of Fuchsian systems of dimension \( 2\times 2\). Derivation of the Painlevé VI equation. (English. Russian original) Zbl 1180.34099 Russ. Math. Surv. 64, No. 1, 45-127 (2009); translation from Usp. Mat. Nauk 64, No. 1, 51-134 (2009). The author discusses an unusual parametrization of the quotient space of the gauge-equivalent linear \(2\times2\) matrix Fuchsian ODEs with singularities at \(\lambda_k\), \(k=0,\dots,n\). The key point of the parametrization is a special triangular normalization of a canonical representative for the class of gauge equivalent ODEs at three distinct points. Namely, if one chooses \(\lambda_0\), \(\lambda_{n-1}\) and \(\lambda_{n}\) as the points of normalization, then the respective residue matrices \(A^{(0)}\), \(A^{(n-1)}\) and \(A^{(n)}\) in the corresponding standard basis are characterized by the conditions \(A^{(0)}_{12}= A^{(n-1)}_{21}=0\) and \(A^{(n)}_{12}=1\) (recall, the conventional normalization implies \(\lambda_0=\infty\) with a diagonal residue matrix \(A^{(0)}\)). As a result, the phase space of a Hamiltonian system defining the Garnier-Painlevé system is naturally constructed as a symplectic algebraic manifold. Adding to this manifold a divisor corresponding to the solutions infinite at this moment yields the whole Okamoto surface of initial conditions. Remarkably, the introduced canonical parametrization of the regular Schlesinger component imposes no restriction on the eigenvalues of the residue matrices \(A^{(i)}\) in spite of the author does not consider the so-called resonant cases.The developed theory is illustrated by the case \(n=3\). With the canonically normalized residue matrices \(A^{(0)}\), \(A^{(2)}\) and \(A^{(3)}\), all the matrix elements become polynomials in the canonical coordinates \(q_1\) and \(p_1\) defined by the entries of \(A^{(1)}\). Using a cross-ratio of \(\lambda_k\) as the time variable and excluding from the corresponding Hamiltonian system \(p_1\), the author reconstructs equation PVI in its traditional rational form. Some different however not obvious choice of the canonical coordinates and the time variable yields an elliptic form of PVI. The paper is concluded by the proof of Okamoto’s theorem that the Painlevé VI field of directions is the only smooth algebraic field of directions on the corresponding extended phase space. Reviewer: Andrei A. Kapaev (St. Petersburg) Cited in 7 Documents MSC: 34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies 33E17 Painlevé-type functions 34A26 Geometric methods in ordinary differential equations 34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms 37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010) 34M03 Linear ordinary differential equations and systems in the complex domain Keywords:Fuchsian system; isomonodromic deformations; Hamiltonian system; phase space; symplectic manifold; Darboux coordinates; Painlevé VI equation; surface of initial data PDFBibTeX XMLCite \textit{M. V. Babich}, Russ. Math. Surv. 64, No. 1, 45--127 (2009; Zbl 1180.34099); translation from Usp. Mat. Nauk 64, No. 1, 51--134 (2009) Full Text: DOI