  Extremal horizons satisfy an equation induced by the Einstein vacuum
equations that determines the shape of the horizon and the manner in which it
rotates (the EEH equation). Until recently, however, the classification of
solutions required the assumption of axial symmetry. Recently, there has been a
breakthrough: Dunajski and Lucietti proved that every non-static solution
possesses a one-dimensional symmetry group. The first part of our work is
inspired by this result. An identity satisfied by the solutions of the EEH
equation has been distilled (Master Identity), which is crucial for studying
their properties. It is a bit stronger than the original Dunajski-Lucietti
identity and leads directly to the rigidity theorem for any value of the
cosmological constant. Master Identity is used for a simple derivation of the
local form of the general static solution of the EEH equation with non-positive
cosmological constant. All the globally defined compact static solutions are
derived. Thus the list of solutions given in the literature is completed. In
the two-dimensional case (which corresponds to horizons in four-dimensional
spacetime), the Einstein-Maxwell equations of an extremal horizon (EMEH) and
the equations of quasi-Einstein spaces are studied. The general solution on a
compact surface with non-zero genus is derived. In the case of zero genus, the
static solutions are investigated and their axial symmetry is proven. Together
with the new results on non-static solutions on sphere of Colling, Katona and
Lucietti that leads to the uniqueness of the
Reissner-Nordstr\"om-(Anti)de-Sitter extremal horizons. Interestingly, the
static rigidity result is also valid for non-compact spaces with a zero first
cohomology group.
