On Transcendental Functions Arising from Integrating Differential Equations in Finite Terms

In this paper, we discuss a version of Galois theory for systems of ordinary differential equations in which there is no fixed list of allowed transcendental operations. We prove a theorem saying that the field of integrals of a system of differential equations is equivalent to the field of rational functions on a hypersurface having a continuous group of birational automorphisms whose dimension coincides with the number of algebraically independent transcendentals introduced by integrating the system. The suggested construction is a development of the algebraic ideas presented by Paul Painlevé in his Stockholm lectures. © 2015, Springer Science+Business Media New York.