Twisting asymptotic symmetries and algebraically special vacuum solutions

In this paper, we study asymptotic symmetries and algebraically special exact solutions in the Newman-Penrose formalism. Removing the hypersurface orthogonal condition in the well studied Newman-Unti gauge, we obtain a generic asymptotic solution space which includes all possible origins of propagating degree of freedom. The asymptotic symmetry of the generalized system extends the Weyl-BMS symmetry by two independent local Lorentz transformations with non-trivial boundary charges, which reveals new boundary degrees of freedom. The generalized Newman-Unti gauge includes algebraically special condition in its most convenient form. Remarkably, the generic solutions satisfying the algebraically special condition truncate in the inverse power of radial expansions and the non-radial Newman-Penrose equations are explicitly solved at any order. Hence, we provide the most general algebraically special solution space and the derivation is self-contained in the Newman-Penrose formalism. The asymptotic symmetry with respect to the algebraically special condition is the standard Weyl-BMS symmetry and the symmetry parameters consist only the integration constant order. We present the Kerr solution and Taub-NUT solution in the generalized Newman-Unti gauge in a simple form.