Classical integrability for beta-ensembles and general Fokker-Planck equations

Understand

Beta-ensembles of random matrices are naturally considered as quantum integrable systems, in particular, due to their relation with conformal field theory, and more recently appeared connection with quantized Painlev\'e Hamiltonians.

• Here we demonstrate that, at least for {\it even integer} beta, these systems are classically integrable, e.g.
• there are Lax pairs associated with them, which we explicitly construct.
• To come to the result, we show that a solution of every Fokker-Planck equation in one space (and one time) dimensions can be considered as a component of an eigenvector of a Lax pair.