On dimension of the space of Killing fields on $k$-symmetric Lorentzian manifolds

We study the Killing equation on $k$-symmetric Lorentzian manifolds. Solutions of this equation form a Lie algebra called the algebra of Killing fields. Our consideration is focused primarily on the dimension of the Lie algebra of Killing fields. The Lorentzian manifolds we consider in this article are the generalized Cahen–Wallach spaces, which are convinient to use because of the coordinate system they have. Using these coordinates, we describe the general solution of the Killing equation on locally indecomposable 2-symmetric Lorentzian manifolds, which are generalized Cahen–Wallach spaces, as was proved by A. S. Galaev and D. V. Alekseevsky. Finally, we give an explicit description of all possible dimensions of the algebra of Killing fields on 2-symmetric Lorentzian manifolds of small dimensions.