(Anti) self-dual Einstein metrics of zero signature, their Petrov classes and connection with Kahler and para-Kahler structures

For (anti) self-dual Einstein metrics, as well as for any (anti) self-dual metrics of zero signature, not $6$ Petrov types are logically possible, but $7$. In addition to the usual types I, D, O, II, III and N the type I$_{0}$ is also possible, described by characteristic root $0$ of multiplicity $4$. A system of anti-self-duality equations for the Riemann tensor is compiled for a metric that is universal in the class of anti-self-dual zero signature metrics. Particular solutions are found for all types except I$_{0}$. We left open the question of the existence of the type I$_{0}$. For an arbitrary metric of zero signature, all almost-Hermitian and almost para-Hermitian structures are found. All Kahler and para-Kahler structures are found for the (anti) self-dual Einstein metric. For a metric of signature $0$, the notion of hyper-Kahler property is introduced for the first time. Its definition differs from the definition of hyper-Kahler Riemannian metrics, but is equivalent to it for dimension $4$. Each (anti) self-dual Einstein metric of zero signature is simultaneously hyper-Kahler and para-hyper-Kahler. Conversely, any hyper-Kahler (para-hyper-Kahler) $4$-metric of zero signature is (anti) self-dual and Einstein metric.