We consider the action principles that are the lower dimensional limits of
the Einstein-Gauss-Bonnet gravity {\it via} the Kaluza-Klein route. We study
the vacua and obtain some exact solutions. We find that the reality condition
of the theories may select one vacuum over the other from the two vacua that
typically arise in Einstein-Gauss-Bonnet gravity. We obtain exact black hole
and cosmological solutions carrying scalar hair, including scalar hairy BTZ
black holes with both mass and angular momentum turned on. We also discuss the
holographic central charges in the asymptotic AdS backgrounds.Comment: Latex, 19 page
