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Monodromy of certain Painlevé-VI transcendents and reflection groups. (English) Zbl 0960.34075

This very extensive paper presents the particular case $\text{PVI}\mu$ of the Painlevé VI equation $\text{PVI}\left(\alpha ,\beta ,\mu ,\delta \right)$, with $\alpha =\frac{{\left(2\mu -1\right)}^{2}}{2}$, $\beta =\gamma =0$, $\delta =\frac{1}{2}$. The single paragraphs of the paper are heading as follows:

1.1 Painlevé VI equation as isomonodromy deformation equation.

1.2 The structure of the analytic continuation.

1.3 Monodromy data and finite-branching solutions of the $\text{PVI}\mu$ equation.

1.4 Monodromy data and reflection groups.

2.0 Global structure of the solutions of Painlevé $\text{VI}\mu$ having critical behaviour of algebraic type.

2.1 Local theory of the solutions of $\text{PVI}\mu$ having critical behaviour of algebraic type.

2.2 The local asymptotic behaviour and the monodromy group of the Fuchsian system.

2.3 From the local asymptotic behaviour to the global one.

2.4 The complete list of algebraic solutions.

MSC:
 34M55 Painlevé and other special equations; classification, hierarchies 34M35 Singularities, monodromy, local behavior of solutions, normal forms