  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.
