  In this paper we consider a Vlasov or collisionless Boltzmann equation
describing the dynamics of planetary rings. We propose a simple physical model,
where the particles of the rings move under the gravitational Newtonian
potential of two primary bodies. We neglect the gravitational forces between
the particles. We use a perturbative technique, which allows to find explicit
solutions at the first order and solutions at the second order, solving a set
of two linear ordinary differential equations.
