Equicontinuity of the parallel flow on compact Brinkmann manifolds

Establish whether the flow generated by the nonzero parallel lightlike vector field on every compact Brinkmann manifold is equicontinuous, equivalently whether its closure in the isometry group is compact.

Background

A Brinkmann manifold is a Lorentz manifold carrying a nonzero parallel lightlike vector field. The paper cites a result establishing geodesic completeness in the compact case and states a conjecture that the flow generated by the parallel field is equicontinuous. This property would imply the existence of a closed geodesic via the paper’s compact-action proposition, but the conjecture is not resolved here.