Dimensional reduction with noncompact transverse spaces has not been explored much because of the tendency to obtain a Kaluza-Klein spectrum running continuously down to zero mass. However, there exist reductions in which the KK continuum begins only above a mass gap, allowing for a localization of gravity in an effective lower dimensional theory. The talk will present such a construction, in which the transverse wave problem is an integrable model. This allows one to analyse the effective theory and learn what life is like with a reduction in which the KK modes nonetheless have a profound impact on the lower dimensional effective theory.