Some non-trivial and non-gradient closed pseudo-Riemannian steady Ricci solitons

In this paper, we study the Ricci soliton equation on compact indecomposable Lorentzian $3$-manifolds that admit a parallel light-like vector field with closed orbits. These compact structures that are geodesically complete, admit non-trivial, i.e., non-Einstein and non-gradient steady Lorentzian Ricci solitons with zero scalar curvature which show the difference between closed Riemannian and pseudo-Riemannian Ricci solitons. The associated potential vector field of a Ricci soliton structure in all the cases that we construct on these manifolds is a space-like vector field. However, we show that there are examples of closed pseudo-Riemannian steady Ricci solitons in the neutral signature $(2,2)$ with zero scalar curvature such that the associated potential vector field can be time-like or null. These compact manifolds are also geodesically complete and they cannot admit a conformal-Killing vector field.