The authors seek to find the generalized static spherically symmetric solutions for vacuum Einstein field equations in \(m+n+2\) dimensions where m and n refer to arbitrary dimensions of Einstein external space and internal space respectively. The global properties of all such space- times are investigated with a view to explore the possibility of spontaneous compactification and existence of regular dimensionally reduced black hole solutions. Apart from the trivial case of the Schwarzschild solution with constant scalar field, all solutions are found to contain naked singularities or else not to be asymptotically flat, as would be expected from the ‘no hair’ theorems.