Abstract
Einstein complex spacetimes admitting null Killing or null homothetic Killing vectors are studied. Such 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 ( $$\mathcal {HH}$$ -spaces) or, if one of the parts of the Weyl tensor vanishes, heavenly spaces ( $$\mathcal {H}$$ -spaces). The explicit complex metrics admitting null Killing vectors are found. Some Lorentzian and ultrahyperbolic slices of these metrics are discussed.
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