Special solutions of the first and second Painlevé equations and singularities of the monodromy data manifold

Abstract

A classification of solutions of the first and second Painleve equations corresponding to a special distribution of poles at infinity is considered. The relation between this distribution and singularities of the two-dimensional complex monodromy data manifold used for the pa- rameterization of the solutions is analyzed. It turns out that solutions of the Painlevee quations have no poles in a certain critical sector of the complex plane if and only if their monodromy data lie in the singularity submanifold. Such solutions belong to the so-called class of "truncated" solutions (integrales tronquee) according to P. Boutroux's classification. It is shown that all known special solutions of the first and second Painleve equations belong to this class.