Homogeneous geodesics of non-unimodular Lorentzian Lie groups and   naturally reductive Lorentzian spaces in dimension three
  We determine, for all three-dimensional non-unimodular Lie groups equipped with a Lorentzian metric, the set of homogeneous geodesics through a point. Together with the results of [C] and [CM2], this leads to the full classification of three-dimensional Lorentzian g.o. spaces and naturally reductive spaces. 
Giovanni Calvaruso and Rosa Anna Marinosci