In this talk, I am going to talk about the construction of stationary axisymmetric black hole solutions to the EV system. These solutions have the property that the spatial support of the matter is a finite, axially symmetric shell located away from the black hole. To this end, I will start by reviewing some of the progress made in the context of solutions to the Einstein-Vlasov system. Then, I will explain how the study of trapped timelike geodesics in a perturbed Kerr spacetime allowed us to provide a one-parameter family of solutions.