A list below contains the problems which I am dealing with nowadays.
- Exact and general solutions of para-Kähler spaces. This issue deals with spaces of the types $[\textrm{any}] \otimes [\textrm{D}]^{nn}$. Especially interesting are Einstein spaces of the types $[\textrm{I}]
\otimes [\textrm{D}]^{nn}$ because no explicit examples of such spaces are known nowadays. Also, new examples of the heavenly spaces of the type $[\textrm{I}] \otimes [\textrm{O}]$ could be found during the analysis of this problem.
- Para-Kähler Einstein geometry of the neutral spaces of the types $[\textrm{D}]\otimes [\textrm{N}]$. Cosmological constant $\Lambda$ is necessarily nonzero in this case. There are two subtypes: $\{ [\textrm{D}]^{nn} \otimes
[\textrm{N}]^{e},[--,++] \}$ and $\{ [\textrm{D}]^{nn} \otimes
[\textrm{N}]^{e},[++,++] \}$. It is supposed that the first of these types could be solved with all the generality. Interesting task is to find examples of the second type.
- Para-Hermite Einstein geometry of the neutral spaces of the types $[\textrm{D}]\otimes [\textrm{N}]$. There are two subtypes: $[\textrm{D}]^{ee} \otimes
[\textrm{N}]^{n}$ and $[\textrm{D}]^{ee} \otimes
[\textrm{N}]^{e}$ with different properties of the congruences of null geodesics. Explicit examples are unknown.
- Spaces equipped with three or four distinct congruences of null strings of the same duality. The metric of a space equipped with three congruences of null strings is known - this is a space of the type $[\textrm{I,II}]^{eee} \otimes
[\textrm{any}]$ (with a special subcase $[\textrm{II}]^{nee} \otimes [\textrm{any}]$). However, the metric of a space of the type $[\textrm{I}]^{eeee} \otimes [\textrm{any}]$ remains unknown. Can the results be specialized to the types $[\textrm{I}]^{eeee} \otimes [\textrm{D}]^{nn}$ and
$[\textrm{I}]^{eeee} \otimes [\textrm{O}]$ (which are para-Kähler)? What types of the traceless Ricci tensor are admitted by such spaces?
- Spaces equipped with four distinct congruences of null geodesics. The problem is to find the metric of a space which is equipped with four distinct shearfree congruences of null geodesics. It is not even known if such a structure exists. If "yes", it is admitted only by spaces of the type $[\textrm{I}]$ with matter. What types of mater are admitted by such spaces?
- Killing vectors in Walker spaces in dimension 4. The main aim is to analyze the Killing symmetries in spaces of the types $[\textrm{deg}]^{n}
\otimes [\textrm{any}]$. Is there an equivalent of the master equation in such spaces?
- Null Killing vectors in spaces with $C_{ab} \ne 0$. It is known (A.C., 2014) that the existence of the null Killing vectors in Einstein spaces implies the existence of the self-dual and anti-self-dual congruences of null strings. Can these results be generalized to the case of non-vacuum spaces.
- Type-$[\textrm{N}] \otimes [\textrm{N}]$ problem - a superposition. It is known (Plebański, Przanowski, Formański, 1998) that a superposition of two heavenly spaces of the types $[\textrm{N}] \otimes [\textrm{O}]$ and
$[\textrm{O}] \otimes [\textrm{N}]$ lead to the space of the type $[\textrm{N}] \otimes
[\textrm{N}]$ with matter. The idea is to consider superposition of two SD spaces of the types $[\textrm{N}] \otimes [\textrm{O}]$ and $[\textrm{O}] \otimes [\textrm{N}]$? The general metric of the SD space of the type
$[\textrm{N}]^{n} \otimes [\textrm{O}]^{n}$ is known. It is desired to find a general form of the metric of the type $[\textrm{N}]^{e} \otimes [\textrm{O}]^{n}$ and then investigate a superposition.
- Type-$[\textrm{N}] \otimes [\textrm{N}]$ spaces with a single symmetry. Is it possible to reduce the vacuum Einstein field equations for the type $[\textrm{N}]^{e} \otimes
[\textrm{N}]^{e}$ with a twist equipped with a single Killing vector to a single equation? Can the counterpart of the Hauser solution be found?
- Type-$[\textrm{N}] \otimes [\textrm{N}]$ spaces with $\Lambda \ne 0$. It is known that the existence of the nonzero cosmological constant simplifies a little type-$[\textrm{N}]$ problem with twist. With $\Lambda \ne 0$ there are only three subtypes, namely $\{ [\textrm{N}]^{e} \otimes
[\textrm{N}]^{e},[--] \}$, $\{ [\textrm{N}]^{e} \otimes
[\textrm{N}]^{e},[+-] \}$ and $\{ [\textrm{N}]^{e} \otimes
[\textrm{N}]^{e},[++] \}$. It is possible to reconstruct known solutions and to find new ones?
Issues 1-4 and 6 deal with neutral geometry (in particular: para-Kähler and para-Hermitian geometry). Issues 8-10 are related to the almost 50-years old problem, namely: to find the second twisting, vacuum type $[\textrm{N}]$ solution. In all these problems a complex methods and spinorial formalism seem to be the best tools for analysis.
A title of the Ph.D. thesis which can be proposed is "Hyperheavenly spaces and their real slices". For realization of this dissertation a basic knowledge of differential geometry, general theory of relativity and partial differential equations is needed. The dissertation can be finalized in 4-5 years.