The study of the canonical forms of Killing tensor in vacuum with Λ

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.