as a perfect fluid and different shapes of the magnetic field can be observed depending on the assumptions on the fluid, like an equation of state or conductivity properties. For example rotating solutions with no poloidal magnetic field can be constructed, cf. [@fr12]. These works can serve as a source of intuition for the study of rotating clouds of Vlasov matter. There is however an important difference. When studying a perfect fluid, the Einstein-Euler system (which describes a space-time containing matter of the type of a perfect fluid) has to be supplemented by an equation of state which captures the physical properties of the fluid under consideration. Depending on the choice of the equation of state, different matter configurations and different electro-magnetic fields can be constructed. For Vlasov matter however there is much less variety in the physical properties of the solutions that can be obtained. The basic assumptions on the particles’ behaviour and how the energy and the angular momentum is distributed among the particles (this is sometimes referred to as a [*microscopic equation of state*]{}) already determines the macroscopic character of the solutions. It turns out that rotating solutions of the EVM-system must have a poloidal magnetic field 