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

Null Killing vectors and geometry of null strings in Einstein spaces