Complete integrability and the Painleve property

Abstract

We present a systematic review of the connection between the complete integrability of (finite-dimensional and infinite-dimensional) Hamiltonian and non-Hamiltonian dynamical systems and the singularity structure of their equations of motion. We make a special attempt to separate the many conjectures in the field from the few rigorous results. The singularity analysis of ordinary and partial differential equations is illustrated by means of four examples. In the Appendix, we present a brief review of the special properties of soliton equations and the relations between them. We summarize these relations in a tabular form.