Type D spaces and quasi-diagonalizability

The coordinate system of Kerr and Debney is used to find the empty Type D metrics with a diverging principal null vector. These spaces are shown to be precisely that subclass of the diverging, empty, algebraically special spaces which are quasi-diagonalizable. This leads to the canonical forms found by Plebanski and Demianski for the empty Type D metrics. These are generalized to a class of charged Type D metrics possessing a cosmological constant. The theory of symmetries in an empty algebraically special space is examined, revealing that those spaces with two commuting Killing vectors are characterized by four real constants, and that if two of these are zero, the space is Type D, and quasi-diagonalizable. The field equations are then linearized, and solved completely. A brief discussion of conformal Killing tensors is given, and an upper bound is found for the number of linearly independent, second order, trace-free, conformal Killing tensors in any Riemannian space of dimension greater than two. Finally, it is shown that the Type D metrics are a natural subclass of those spaces with quasi-diagonal metrics and non-redundant, conformal Killing tensors