Integrable Poisson algebras and two-dimensional manifolds

Abstract

The relations between integrable Poisson algebras with three generators and two-dimensional symplectic manifolds are investigated. It is shown that for a given integrable Poisson algebra there exists a two-dimensional symplectic manifold such that the Poisson algebra generated by the coordinates of M coincides with the algebra . Vice versa the coordinates of a given smooth two-dimensional symplectic manifold M embedded in generate an integrable Poisson algebra. Moreover, smooth Poisson algebraic maps between two integrable Poisson algebras are governed by equations involving the symplectic manifolds corresponding to these algebras.