The separable analogue of Kerr in Newtonian gravity

General Relativity's Kerr metric is famous for its many symmetries which are responsible for the separability of the Hamilton-Jacobi equation governing the geodesic motion and of the Teukolsky equation for wave dynamics. We show that there is a unique stationary and axisymmetric Newtonian gravitational potential that has exactly the same dual property of separable point-particle and wave motion equations. This `Kerr metric analogue' of Newtonian gravity is none other than Euler's 18th century problem of two-fixed gravitating centers.