On conformal factor in the conformal Killing equation on the $2$-symmetric five-dimensional indecomposable Lorentzian manifold

Conformally Killing vector fields are a natural generalization of Killing vector fields and play an important role in the study of the group of conformal transformations of a manifold, Ricci flows on a manifold, and the theory of Ricci solitons. Pseudo-Riemannian symmetric spaces of order $k$, where $k \geq 2$, arise in the study of pseudo-Riemannian geometry and in physics. At present, they have been investigated in cases $k=2, 3$ by D. V. Alekseevsky, A. S. Galaev and others. In the case of low dimensions, these spaces and Killing vector fields on them were studied by D. N. Oskorbin, E. D. Rodionov, and I. V. Ernst. Ricci solitons are a generalization of Einstein's metrics on (pseudo) Riemannian manifolds, and their equation has been studied on various classes of manifolds by many mathematicians. In particular, D. N. Oskorbin and E. D. Rodionov found a general solution of the Ricci soliton equation on $2$-symmetric Lorentzian manifolds of low dimension, and proved the local solvability of this equation in the class of $3$-symmetric Lorentzian manifolds. For a single Einstein constant in the Ricci soliton equation the Killing vector fields make it possible to find the general solution of the Ricci soliton equation corresponding to the given constant. However, for different values of the Einstein constant, conformally Killing vector fields play the role of Killing fields. Therefore, there is a need to study them. In this paper, we investigate the conformal analogue of the Killing equation on five-dimensional $2$-symmetric indecomposable Lorentzian manifolds, and investigate the properties of the conformal factor of the conformal analogue of the Killing equation on them. Nontrivial examples of conformally Killing vector fields with a variable conformal factor are constructed.