  Scalar, vector and tensor perturbations on the Kerr spacetime are governed by
equations that can be solved by separation of variables, but the same is not
true in generic stationary and axisymmetric geometries. This complicates the
calculation of black-hole quasi-normal mode frequencies in theories that
extend/modify general relativity, because one generally has to calculate the
eigenvalue spectrum of a two-dimensional partial differential equation (in the
radial and angular variables) instead of an ordinary differential equation (in
the radial variable). In this work, we show that if the background geometry is
close to the Kerr one, the problem considerably simplifies. One can indeed
compute the quasi-normal mode frequencies, at least at leading order in the
deviation from Kerr, by solving an ordinary differential equation subject to
suitable boundary conditions. Although our method is general, in this paper we
apply it to scalar perturbations on top of a Kerr black hole with an anomalous
quadrupole moment, or on top of a slowly rotating Kerr background.
