  Observations of clusters of galaxies show ubiquitous presence of X-ray
cavities, presumably blown by the AGN jets. We consider magnetic field
structures of these cavities. Stability requires that they contain both
toroidal and poloidal magnetic fields, while realistic configurations should
have vanishing magnetic field on the boundary. For axisymmetric configurations
embedded in unmagnetized plasma, the continuity of poloidal and toroidal
magnetic field components on the surface of the bubble then requires solving
the elliptical Grad-Shafranov equation with both Dirichlet and Neumann boundary
conditions. This leads to a double eigenvalue problem, relating the pressure
gradients and the toroidal magnetic field to the radius of the bubble. We have
found fully analytical stable solutions. This result is confirmed by numerical
simulation. We present synthetic X-ray images and synchrotron emission profiles
and evaluate the rotation measure for radiation traversing the bubble.
