The general structure of nonexpanding hyperheavenly spaces is presented. We find conformal Killing equations and their integrability conditions in spinorial formalism. The reduction of Killing equations to one master equation is also presented. We generalize the Killing problem on the case of a nonzero cosmological constant and a conformal Killing factor. Finally, an example of the complex [N]⊗[N] space with the conformal Killing vector is considered and the respective metric is explicitly given.

The notion of a weak hyperheavenly space is introduced and the general form of the metric for such a space is found. Spinorial connection and curvature are also calculated. It is shown that any four-dimensional Walker space is a weak nonexpanding real -space with the metric of the signature . The metrics of the self-dual Walker and the self-dual Einstein–Walker spaces are given. The general form of the metric for any Walker space admitting both self-dual and anti-self-dual totally null parallel 2-distributions is found.

Spinorial formalism employed in the weak hyperheavenly space theory is used to study Osserman spaces. General forms of metrics of signature (+ + − −) for self-dual pointwise Osserman, globally Osserman and globally Jordan–Osserman spaces admitting a two-dimensional totally null self-dual integrable distribution are found by searching for solutions to respective hyperheavenly equations with Λ. As is shown the metrics obtained describe locally all self-dual pointwise Osserman, globally Osserman and globally Jordan–Osserman spaces of types II, III, N, 0 and D (where in the case of type D one assumes that the multiple undotted Penrose spinors are real).

All Killing symmetries in complex H-spaces with Λ in terms of the Plebański -Robinson -Finley coordinate system are found. All H-metrics with Λ admitting a null Killing vector are explicitly given. It is shown that the problem of non-null Killing vector reduces to looking for solution of the Boyer -Finley -Plebański (Toda field) equation.