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 $f(T)$ 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 $f(T)$ 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.

共形对称性可作为通过共形矢量场 (CVF) 研究和分类爱因斯坦场方程 (EFE) 精确解的来源。众所周知，这种分类会导致一类重要的对称性，称为 Killing 对称性，它是产生物理守恒定律的来源。在本文中，我们首先探讨了各种类型的 Kantowski-Sachs (KS) 和 Bianchi 类型 III 解决方案$F(吨)$重力通过采用一些代数技术。利用上述技术，我们知道存在 30 个 KS 和 Bianchi 类型 III 时空允许解$F(吨)$重力。通过精确检查所有类，我们熟悉了 25 个解决方案制定了非保形平坦度量，而 5 个解决方案的其余部分倾向于制定保形平坦度量。我们利用由此产生的解决方案通过直接集成方法找到 CVF。经过详细研究，我们发现在两种情况下，时空允许适当的 CVF，而在其他情况下，时空要么变得共形平坦，要么允许相似向量场 (HVF) 或杀死向量场 (KVF) ）。所考虑的时空的 CVF 的整体维度已经证明是四、五、六或十五。