A Directive for obtaining (Algebraically) General solutions in General Relativity based on the Canonical Killing tensor forms

This work follows earlier investigations in which the existence of canonical Killing tensor forms and the application of general null tetrad transformations led to a variety of solutions (Petrov types D, III, N) in vacuum with a cosmological constant. {Among those, a distinct Petrov type D family was extracted and characterized by a topological product of two-dimensional constant-curvature spaces admitting the canonical form K²μ ν}. This is a general family of spacetimes with constant curvature and it is derived and presented here in full detail. In addition, an algebraically general solution exhibiting {the exact same non-zero spin coefficients is introduced. Beyond this, we introduce an algebraically general solution (Petrov type I), obtained by imposing the same canonical Killing tensor form and applying a Lorentz transformation within the (anti-)symmetric null tetrad transformation. The resulting geometry describes a non-stationary, cylindrically symmetric spacetime in vacuum with cosmological constant}. On this basis, we propose a new directive: by assuming the canonical forms of Killing tensors and implementing Lorentz transformations within the (anti-)symmetric null tetrad concept, a broader class of (algebraically) general solutions of Einstein’s equations can be derived.