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Abstract

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. 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, 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.