  We show how all known N=2, d=4,5,6 maximally supersymmetric vacua (Hpp-waves
and aDSxS solutions) are related through dimensional reduction/oxidation
preserving all the unbroken supersymmetries. In particular we show how the N=2,
d=5 family of vacua (which are the near-horizon geometry of supersymmetric
rotating black holes) interpolates between aDS_2xS^3 and aDS_3xS^2 in parameter
space and how it can be dimensionally reduced to an N=2, d=4 dyonic
Robinson-Bertotti solution with geometry aDS_2xS^2 and oxidized to an N=2, d=6
solution with aDS_3xS^3 geometry (which is the near-horizon of the self-dual
string).
