  We prove that, contrary to the situation with time-like and space-like
parallel vector fields, there are real gravitational fields satisfying
Einsteins equations of gravity and admitting nontrivial light-like parallel
vector fields; we solve completely the field equations under this geometric
constraint and obtain through Bianchi identities an interesting property for
the Riemann curvature tensor corresponding with a real gravitational field
admitting a nontrivial parallel vector field. The solutions we obtained turn
out to be generically wave-like and then we prove by a geometric method that
this class of solutions is quite different from the class of solutions
corresponding with spherical waves.
