  HH-spaces, i.e., complex spacetimes, of Petrov type NxN are determined by a
trio of pde's for two functions, lambda and a, of three independent variables
(and also two gauge functions, chosen to be two of the independent variables if
one prefers). As in common integrable systems, these form a second order,
linear system for lambda; howver, here the integrability conditions, involving
a, are more complicated than is common. Therefore, with the hope of finding new
solutions, these equations are now constrained to also admit both one and two
homothetic or Killing vectors. The case with one Killing and one homothetic
vector reduces these equations to two ode's for two unknown functions of the
one remaining variable.
  In addition, we also describe in detail the explicit forms of the metric,
tetrad, connections, and curvature for twisting HH-spaces of Petrov type NxN,
modulo the determining equations. This simplifies considerably the process of
obtaining these details cleanly from earlier articles on the subject, thus
simplifying access to the research area.

General Relativity and Quantum Cosmology