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