The Connection between Partial Differential Equations Soluble by Inverse Scattering and Ordinary Differential Equations of Painlevé Type

A completely integrable partial differential equation is one which has a Lax representation, or, more precisely, can be solved via a linear integral equation of Gel’fand–Levitan type, the classic example being the Korteweg–de Vries equation. An ordinary differential equation is of Painlevé type if the only singularities of its solutions in the complex plane are poles. It is shown that, under certain restrictions, if G is an analytic, regular symmetry group of a completely integrable partial differential equation, then the reduced ordinary differential equation for the G-invariant solutions is necessarily of Painlevé type. This gives a useful necessary condition for complete integrability, which is applied to investigate the integrability of certain generalizations of the Korteweg–de Vries equation, Klein–Gordon equations, some model nonlinear wave equations of Whitham and Benjamin, and the BBM equation.