This book, now in its second edition, introduces the singularity analysis of differential and difference equations via the Painlev test and shows how Painlev analysis provides a powerful algorithmic approach to building explicit solutions to nonlinear ordinary and partial differential equations. It is illustrated with integrable equations such as the nonlinear Schr dinger equation, the Korteweg-de Vries equation, H non-Heiles type Hamiltonians,...