  All axisymmetric solutions to the near-horizon geometry equation with a
cosmological constant defined on a topological $2$-sphere were derived. The
regularity conditions preventing cone singularity at the poles were accounted
for. The one-to-one correspondence of the solutions with the extremal horizons
in the Kerr-(anti-)de Sitter spacetimes was found. A solution corresponding to
the triply degenerate horizon was identified and characterized. The solutions
were also identified among the solutions to the Petrov type D equation.
