Specificity of Petrov classification of (anti-)self-dual zero signature metrics

A.Z. Petrov divided 4-metrics of the zero signature into 6 types, which later began to be denoted I, D, O, II, N, III. However, in the case of (anti)-self-duality, the $\lambda$-matrix, on the basis of which Petrov built his classification, acquires specificity. First, the determinant of this $\lambda$-matrix has a root 0 of multiplicity at least 3. Secondly, the multiplicity of this root cannot be 5. These two circumstances lead to the fact that there are not 6, but 7 different types of metrics. A new type I$_{0}$ appears, whose characteristic number 0 has multiplicity 4. This type does not coincide with I, since for type I the multiplicity of the root 0 is three. Examples of metrics, expressed in terms of elementary functions, of all seven types are constructed.