Specificity of Petrov Classification of (Anti-)Self-Dual Zero Signature Metrics

A.Z. Petrov divided 4-metrics of signature 0 into 6 types, which later began to be denoted by I , D , O , II , N , III . However, in the case of (anti-)self-duality, the -matrix, on the basis of which Petrov built his classification, acquires specificity. First, the determinant of this -matrix has a root 0 of multiplicity at least 3. Second, 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 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 7 types are constructed.