Classification of algebraic solutions of irregular Garnier systems

Abstract

We prove that algebraic solutions of Garnier systems in the irregular case are of two types. The classical ones come from isomonodromic deformations of linear equations with diagonal or dihedral differential Galois group; we give a complete list in the rank-2 case (two indeterminates). The pull-back ones come from deformations of coverings over a fixed degenerate hypergeometric equation; we provide a complete list when the differential Galois group is $\text{SL}_{2}(\mathbb{C})$. As a byproduct, we obtain a complete list of algebraic solutions for the rank-2 irregular Garnier systems.