Painleve VI, Rigid Tops and Reflection Equation

Abstract

We show that the Painlev{\'e} VI equation has an equivalent form of the non-autonomous Zhukovsky-Volterra gyrostat. This system is a generalization of the Euler top in $C^3$ and include the additional constant gyrostat momentum. The quantization of its autonomous version is achieved by the reflection equation. The corresponding quadratic algebra generalizes the Sklyanin algebra. As by product we define integrable XYZ spin chain on a finite lattice with new boundary conditions.