  We prove that the universal covering of a complete locally symmetric normal
metric contact pair manifold is a Calabi-Eckmann manifold. Moreover we show
that a complete, simply connected, normal metric contact pair manifold such
that the foliation induced by the vertical subbundle is regular and reflections
in the integral submanifolds of the vertical subbundle are isometries, then the
manifold is the product of globally $\phi$-symmetric spaces and fibers over a
locally symmetric space endowed with a symplectic pair.
