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Complex and real, vacuum spaces with both self-dual and anti-self-dual partsof the Weyl tensor being of the type [N] are considered. Such spaces areclassified according to two criteria. The first one takes into account theproperties of the congruences of totally null, geodesic 2-dimensional surfaces(the null strings). The second criterion is based on investigations of theproperties of the intersection of these congruences. It is proved that thereexist six distinct types of the $[\textrm{N}] \otimes [\textrm{N}]$ spaces. Newexamples of the Lorentzian slices of the complex metrics are presented. Also,some type $[\textrm{N}] \otimes [\textrm{N}]$ spaces which do not possesLorentzian slices are considered.

Classification of complex and real, vacuum spaces of the type $[\textrm{N}] \otimes [\textrm{N}]$