  We determine all infinitesimal transverse deformations of extreme horizons in
Einstein-Maxwell theory that preserve axisymmetry. In particular, we show that
the general static transverse deformation of the AdS(2) X S(2) near-horizon
geometry is a two-parameter family, which contains the known extreme charged,
accelerating, static black hole solution held in equilibrium by an external
electric or magnetic field (Ernst solution) and a special case of the extreme
Kerr-Newman-Melvin solution. More generally, we find a three-parameter family
of deformations of the extreme Kerr-Newman horizon, which contains the extreme
Kerr-Newman-Melvin solution and a rotating generalisation of the Ernst
solution. We also consider vacuum gravity with a cosmological constant and
prove uniqueness of axisymmetric transverse deformations of the extreme
Kerr-AdS horizon. Finally, we completely classify transverse deformations of
extreme horizons in three-dimensional Einstein-Maxwell theory with a negative
cosmological constant.
