Multidimensional theories still remain attractive from the point of view of better understanding of fundamental interactions. In this paper we consider a six - dimensional Kaluza -- Klein type model at the classical level. We derive static spherically symmetric solutions to the multidimensional Einstein equations. They are fundamentally different from four - dimensional Schwarzschild solutions: they are horizon free and the presence of massless dilaton field has the same dynamical effect as the existence of additional massive matter in the system. Then we analyse the motion of test particles in such spherically symmetric configurations. The emphasis is put on some observable quantities like redshifts. It is suggested that strange features of emission lines from active galactic nuclei as well as quasar - galaxy associations may in fact be manifestations of multidimensionality of our world.