Painlevé test and partial differential equations

The Painlevé property for ordinary differential equations, namley that all movable singularities in the solutions are poles, has been known since the famous work of Sonya Kovalevskaya, to be related to integrability. In recent years, it has been generalized so as to apply to (nonlinear) partial differential equations. The conjecture is that a partial differential equation is integrable (by the inverse scattering transform or Bācklund transform, or some other method), if and only if it possesses the Painlevé property possibly after a change of variables (see Steeb and Euler 1988)…