Assuming the most general form of static spherically symmetric space-times, we search for the conformal vector fields in [Formula: see text] gravity by means of algebraic and direct integration approaches. In this study, there exist six cases which on account of further study yield conformal vector fields of dimension four, six and fifteen. During this study, we also recovered some well-known static spherically symmetric metrics announced in the current literature.

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In this paper, first, we deduce various classes of Bianchi type-I perfect fluid solutions in the [Formula: see text] gravity. In order to obtain the above-mentioned classes, we use some algebraic techniques that help to formulate 18 cases. As an application, we bifurcate the obtained classes according to their conformal vector fields (CVFs). Inspecting each class using the process of direct integration, we come to know that in one case, the space-time admits proper CVFs whereas in rest of the cases, obtained metrics either become conformally flat or admit underlying symmetries of the CVFs i.e. homothetic vector fields (HVFs) or Killing vector fields (KVFs). The overall dimension of CVFs for the Bianchi type-I perfect fluid space-times in [Formula: see text] gravity turns out to be three, four, five, six and fifteen.

Conformal symmetries act as a source to investigate and classify exact solutions of the Einstein field equations (EFEs) via conformal vector fields (CVFs). It is well known that such classification leads to an important class of symmetries known as Killing symmetry which is the source of generating conservation laws of physics. In this paper, first we explore various classes of Kantowski–Sachs (KS) and Bianchi type III solutions in [Formula: see text] gravity by adopting some algebraic techniques. Utilizing the above-mentioned technique, we come to know that there exist 30 cases where the KS and Bianchi type III space-times admit solutions in [Formula: see text] gravity. Inspecting all the classes precisely, we familiarize that 25 solutions formulate non-conformally flat metrics whereas rest of the five solutions tend to formulate conformally flat metrics. We utilize the resulting solutions in finding the CVFs via direct integration approach. After a detailed study, we found that in two cases, the space-times admit proper CVFs, whereas in rest of the cases, the space-times either become conformally flat or it admit homothetic vector fields (HVFs) or Killing vector fields (KVFs). The overall dimension of CVFs for the space-times under consideration has turned out to be four, five, six or fifteen.

The purpose of this paper is to explore the anisotropic exact Bianchi type II solutions in [Formula: see text] gravity, where [Formula: see text] denotes the torsion scalar. We utilize the solutions to discuss conformal vector fields (CVFs) and energy conditions. In the first slot of this study, we find the CVFs. The CVFs being a generalization of the Killing vector fields (KVFs) are affiliated with the conservation laws of physics. Corresponding to the obtained solutions, we observe law of conservation of (linear or generalized) momentum. In the second slot, we derive the constraints under which the solution classes can admit certain energy conditions.