Complex and real vacuum spaces with both self-dual and anti-self-dual parts of the Weyl tensor being of the type [N] are considered. Such spaces are classified according to two criteria. The first one takes into account the properties of the congruences of totally null geodesic 2-dimensional surfaces (the null strings). The second criterion is the properties of the intersection of these congruences. It is proved that there exist six distinct types of the [N] ⊗ [N] spaces. New examples of the Lorentzian slices of the complex metrics are presented. Some types of [N] ⊗ [N] spaces which do not possess Lorentzian slices are also considered.
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