Symmetry, Singularity and Integrability: The final question?

Abstract

We explore the relationship between the integrability of ordinary differential equation in terms of analytic functions and the possession of symmetries. By way of example we see that the relationship as traditionally perceived is quite inadequate. We propose a more complete hypothesis and speculate whether this provides the conclusive statement of the relationship between symmetry and integrability. The hypothesis is that an ordinary differential equation integrable in terms of analytic functions is related by some transformation, generally nonlocal, to a linear equation.