  We review the six dimensional universal extra dimension models compactified
on the sphere $S^2$, the orbifold $S^2/Z_2$, and the projective sphere, which
are based on the spontaneous compactification mechanism on the sphere. In
particular, we spell out the application of the Newman-Penrose eth-formalism on
these models with some technical details on the derivation of the Kaluza-Klein
modes and their interactions, and revisit the problem in the existence of the
zero mode of $U(1)_X$ additional gauge boson required for the spontaneous
compactification. We also explain the theoretical background on the vacuum
stability argument for the upper bound on the ultraviolet cutoff scale.
