This paper is the initial part of a comprehensive study of spacetimes that admit the canonical forms of Killing tensor in General Relativity. The general scope of the study is to derive either new exact solutions of Einstein’s equations that exhibit hidden symmetries or to identify the hidden symmetries in already known spacetimes that may emerge during the resolution process. In this preliminary paper, we first introduce the canonical forms of Killing tensor, based on a geometrical approach to classify the canonical forms of symmetric 2-rank tensors, as postulated by R. V. Churchill. Subsequently, the derived integrability conditions of the canonical forms serve as additional equations transforming the under-determined system of equations, comprising of Einstein’s Field Equations and the Bianchi Identities (in vacuum with \(\Lambda \)), into an over-determined one. Using a null rotation around the null tetrad frame we manage to simplify the system of equations to the point where the geometric characterization (Petrov Classification) of the extracted solutions can be performed and their null congruences can be characterized geometrically. Therein, we obtain multiple special algebraic solutions according to the Petrov classification (D, III, N, O) where some of them appeared to be new. The latter becomes possible since our analysis is embodied with the usage of the Newman-Penrose formalism of null tetrads.

本文是对时空进行全面研究的最初部分，该时空承认广义相对论中 Killing 张量的规范形式。该研究的一般范围是推导出表现出隐藏对称性的爱因斯坦方程的新精确解，或者识别在解析过程中可能出现的已知时空中的隐藏对称性。在这篇初步论文中，我们首先介绍了 Killing 张量的经典形式，它基于几何方法对对称 2 秩张量的经典形式进行分类，如 R. V. Churchill 所假设的那样。随后，规范形式的推导可积性条件作为附加方程，将由爱因斯坦场方程和比安奇恒等式（在真空中为 \（\Lambda \））组成的欠定方程组转换为超定方程组。使用围绕零四元组框架的零旋转，我们设法将方程组简化到可以执行提取解的几何表征（彼得罗夫分类）并且可以几何表征它们的零同余的程度。其中，我们根据彼得罗夫分类 （D， III， N， O） 获得了多个特殊代数解，其中一些似乎是新的。后者成为可能，因为我们的分析体现在使用零四元的 Newman-Penrose 形式。