Painlev e-Calogero correspondence

The so called “Painlevé-Calogero correspondence” relates the sixth Painlevé equation with an integrable system of the Calogero type. This relation was recently generaized to the other Painlevé equations and a “multi-component” analogue. This paper reviews these results. 1 Historical background It was at the beginning of the twentieth century that Painlevé discovered what are nowadays called the “Painlevé equations” [1]. Painlevé obtained those equations in the course of classification of second order nonlinear algebraic ordinary differential equations “without movable critical point”. The classification was eventually completed by his student Gambier[2], who supplemented several cases (in particular, the sixth equation) that Painlevé overlooked. The property that the differential equation be free of movable critical point, which lies in the heart of Painlevé’s work, is now called the “Painlevé property”. This kind of analysis is generally referred to as “Painlevé analysis”. Actually, a prototype of Painlevé’s method can be found in Kowalevskaya’s work on integrability of the motion of a rigid body [3]. In this respect, this method should be rather called “Kowalevskaya-Painlevé analysis”. At the time when Painlevé’s classification was being completed, R. Fuchs (son of L. Ruchs, who’s name is coined in the notion of “Fuchsian differential equations”, “Fuchsian groups”, etc.) proposed two new approaches to the sixth Painlevé equations [4]: