An Isomonodromy Cluster of Two Regular Singularities

We consider a linear 2 × 2 matrix ODE with two coalescing regular singularities. This coalescence is restricted with an isomonodromy condition with respect to the distance between the merging singularities in a way consistent with the ODE. In particular, a zero-distance limit for the ODE exists. The monodromy group of the limiting ODE is calculated in terms of the original one. This coalescing process generates a limit for the corresponding nonlinear systems of isomonodromy deformations. In our main example the latter limit reads as P6 → P5, where Pn is the n-th Painlevé equation. We also discuss some general problems which arise while studying the above-mentioned limits for the Painlevé equations. 2000 Mathematics Subject Classification: 34M55, 33E17, 33E30 Short title: Isomonodromy Cluster