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We analyze a class of physical properties, forming the content of theso-called von Zeipel theorem, which characterizes stationary, axisymmetric,non-selfgravitating perfect fluids in circular motion in the gravitationalfield of a compact object. We consider the extension of the theorem to themagnetohydrodynamic regime, under the assumption of an infinitely conductivefluid, both in the Newtonian and in the relativistic framework. When themagnetic field is toroidal, the conditions required by the theorem areequivalent to integrability conditions, as it is the case for purelyhydrodynamic flows. When the magnetic field is poloidal, the analysis for therelativistic regime is substantially different with respect to the Newtoniancase and additional constraints, in the form of PDEs, must be imposed on themagnetic field in order to guarantee that the angular velocity $\Omega$ dependsonly on the specific angular momentum $\ell$. In order to deduce such physicalconstraints, it is crucial to adopt special coordinates, which are adapted tothe $\Omega={\rm const}$ surfaces. The physical significance of these resultsis briefly discussed.

Von Zeipel's theorem for a magnetized circular flow around a compact object