Null Killing vectors and geometry of null strings in Einstein spaces

Einstein complex spacetimes admitting null Killing or null homothetic Killing vectors are studied. These vectors define totally null and geodesic 2-surfaces called the null strings or twistor surfaces. Geometric properties of these null strings are discussed. It is shown, that spaces considered are hyperheavenly spaces (HH-spaces) or, if one of the parts of the Weyl tensor vanishes, heavenly spaces (H-spaces). The explicit complex metrics admitting null Killing vectors are found. Some Lorentzian and ultrahyperbolic slices of these metrics are discussed

Classification of the Killing Vectors in Nonexpanding HH-Spaces with Lambda

Conformal Killing equations and their integrability conditions for nonexpanding hyperheavenly spaces with Lambda are studied. Reduction of ten Killing equations to one master equation is presented. Classification of homothetic and isometric Killing vectors in nonexpanding hyperheavenly spaces with Lambda and homothetic Killing vectors in heavenly spaces is given. Some nonexpanding complex metrics of types [III,N]x[N] are found. A simple example of Lorentzian real slice of the type [N]x[N] is explicitly given

Hyperheavenly spaces and their application in Walker and para-K\"ahler geometries: part II

4-dimensional spaces equipped with congruences of null strings are considered. It is assumed that a space admits a congruence of expanding self-dual null strings and its self-dual part of the Weyl tensor is algebraically degenerate. Different Petrov-Penrose types of such spaces are analyzed. A special attention is paid to para-K\"ahler Einstein spaces. All para-K\"ahler Einstein metrics of spaces with algebraically degenerate self-dual Weyl spinor are found in all the generality

Proper conformal symmetries in SD Einstein spaces

Proper conformal symmetries in self-dual (SD) Einstein spaces are considered. It is shown, that such symmetries are admitted only by the Einstein spaces of the type [N]x[N]. Spaces of the type [N]x[-] are considered in details. Existence of the proper conformal Killing vector implies existence of the isometric, covariantly constant and null Killing vector. It is shown, that there are two classes of [N]x[-]-metrics admitting proper conformal symmetry. They can be distinguished by analysis of the associated anti-self-dual (ASD) null strings. Both classes are analyzed in details. The problem is reduced to single linear PDE. Some general and special solutions of this PDE are presented