A Directive for Obtaining Algebraically General Solutions of Einstein Equations Based on the Canonical Killing Tensor Forms

This work serves as a sequel to our previous study, where, by assuming the existence of canonical Killing tensor forms and applying a general null tetrad transformation, we obtained a variety of solutions (Petrov types D, III, N) in vacuum with cosmological constant $\Lambda$. Among these, there is a unique Petrov type D solution with a shear-free, diverging, and non-geodesic null congruence that admits the $K^2_{\mu \nu}$ canonical form, which we present in full detail. Additionally, we introduce a Petrov type I solution with a shear-free, diverging, and non-geodesic null congruence, obtained by starting from the same canonical form and employing Lorentz transformations within the concept of symmetric null tetrads, instead of a general null tetrad transformation. Building on this and following the concept of symmetric null tetrads, we propose a new directive. This suggests that, by assuming the canonical forms of the Killing tensor and employing Lorentz transformations that correlate the spin coefficients among themselves, such as $\pi=-\bar{\tau}$ and $\kappa = -\bar{\nu}$, one can obtain a broader class of algebraically general solutions to Einstein's equations, rather than relying solely on boosts and spatial rotations.