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Nonintegrability criteria for a class of differential equations with two regular singular points

Criteria for nonintegrability (in the sense of inexistence of continuous, single-valued first integrals in the complex domain) are given for first-order differential equations whose linear part is a homogeneous equation with several regular singular points (in a bounded domain of the complex plane). The method used is the poly-Painlevé test. Local equivalence maps between the nonlinear equations and their linear part are used in the proofs, and the (noncommutative) group of monodromy maps is studied to establish dense branching of solutions, hence nonintegrability. A dicussion concerning the notion of first integral is included.