Painlevé Solutions and Algebraic Complex Integrability

Abstract

This chapter discusses some basic consequences of the definition of algebraic complete integrability (ACI). The ACI systems are integrable systems whose trajectories are straight-line motions on complex algebraic tori. Ultimately, the solutions to such problems can be expressed in terms of theta functions and quadratures involving Abelian integrals. Most well-known integrable systems enjoy these properties. In the recent years, much progress has been made in solving such systems using ingenious techniques involving Lie algebras, infinite dimensional Grassmanians, and various tools of algebraic geometry. The Lie algebras method and infinite dimensional Grassmanians method have not addressed the question of what to do with a specific system or class of systems by some physicist who wants to know if any elements of the class are algebraic complete integrable, and, if so, how does one solve and understand the geometry of these systems in a straightforward way.