  Gravitational waves with parallel rays are known to have remarkable
properties: Their orbit space of null rays possesses the structure of a
non-relativistic spacetime of codimension-one. Their geodesics are in
one-to-one correspondence with dynamical trajectories of a non-relativistic
system. Similarly, the null dimensional reduction of Klein-Gordon's equation on
this class of gravitational waves leads to a Schroedinger equation on curved
space. These properties are generalized to the class of gravitational waves
with a null Killing vector field, of which we propose a new geometric
definition, as conformally equivalent to the previous class and such that the
Killing vector field is preserved. This definition is instrumental for
performing this generalization, as well as various applications. In particular,
results on geodesic completeness are extended in a similar way. Moreover, the
classification of the subclass with constant scalar invariants is investigated.
