This paper considers the relaxation of a smooth two--dimensional vortex to axisymmetry after the application of an instantaneous, weak external strain field. In this limit the disturbance decays exponentially in time at a rate that is linked to a pole of the associated linear inviscid problem (known as a Landau pole). As a model of a typical vortex distribution that can give rise to cat's eyes, here distributions are examined that have a basic Gaussian shape but whose profiles have been artificially flattened about some radius r_c. A numerical study of the Landau poles for this family of vortices shows that as r_c$ is varied so the decay rate of the disturbance moves smoothly between poles as the decay rates of two Landau poles cross. Cat's eyes that occur in the nonlinear evolution of a vortex lead to an axisymmetric azimuthally averaged profile with an annulus of approximately uniform vorticity, rather like the artificially flattened profiles investigated. Based on the stability of such profiles it is found that finite thickness cat's eyes can persist (i.e. the mean profile has a neutral mode) at two distinct radii, and in the limit of a thin flattened region the result that vanishingly thin cat's eyes only persist at a single radius is recovered. The decay of non--axisymmetric perturbations to these flattened profiles for larger times is investigated and a comparison made with the result for a Gaussian profile.