  We consider restrictions placed by geodesic completeness on spacetimes
possessing a null parallel vector field, the so-called Brinkmann spacetimes.
This class of spacetimes includes important idealized gravitational wave models
in General Relativity, namely the plane-fronted waves with parallel rays, or
pp-waves, which in turn have been intensely and fruitfully studied in the
mathematical and physical literatures for over half a century. More concretely,
we prove a restricted version of a conjectural analogue for Brinkmann
spacetimes of a rigidity result obtained by M.T. Anderson for stationary
spacetimes. We also highlight its relation with a long-standing 1962 conjecture
by Ehlers and Kundt. Indeed, it turns out that the subclass of Brinkmann
spacetimes we consider in our main theorem is enough to settle an important
special case of the Ehlers-Kundt conjecture in terms of the well known class of
Cahen-Wallach spaces.
