Exact solution of the sixth Painlevé equation and exotic asymptotic expansion (Q378191)

The Painlevé VI equation is the following ordinary differential equation \[ \begin{multlined} y''=\frac{yx^{2}}{2}\left(\frac{1}{y}+\frac{1}{y-1}+\frac{1}{y-x}\right)-y'\left(\frac{1}{x}+\frac{1}{x-1}+\frac{1}{y-x}\right)\\ +\frac{y(y-1)(y-x)} {x^2(x-1)^2}\left(a+b\frac{x}{y^2}+c\frac{x-1}{(y-1)^2}+d\frac{x(x-1)}{(y-x)^2}\right).\end{multlined} \] In the paper [\textit{A. D. Bryuno} and \textit{I. V. Goryuchkina}, Trans. Mosc. Math. Soc. 2010, 1--104 (2010); translation from Tr. Mosk. Mat. O.-va. 71, 6--118 (2010; Zbl 1215.34113)] the authors found all asymptotic expansion of solutions near a singular point \(x=0\). For some special values of the parameters \(a,b,c,d\) expressions for the solutions can be found. In particular, Picard and Painlevé found general integrals in the case \(a=b=c=d-\frac{1}{2}=0\). Solutions that are obtained by the application of the automorphism group are called Picard-type solutions. In the paper under review, in the case \(a=b=c=d-\frac{1}{2}=0\), the asymptotic expansions of the Picard solutions are compared with the formal expansions obtained in paper mentioned above. It is proved that all formal asymptotic forms of solutions are associated with a Picard-type solution.