  A d-dimensional spacetime is "axisymmetric" if it possesses an SO(d-2)
isometry group whose orbits are (d-3)-spheres. In this paper, algebraically
special, axisymmetric solutions of the higher dimensional vacuum Einstein
equation (with cosmological constant) are investigated. Necessary and
sufficient conditions for static axisymmetric solutions to belong to different
algebraic classes are presented. Then general (possibly time-dependent)
axisymmetric solutions are discussed. All axisymmetric solutions of algebraic
types II, D, III and N are obtained.
