Null Killing vectors and geometry of null strings in Einstein spaces

Einstein complex spacetimes admitting null Killing or null homothetic Killing vectors are studied. These vectors define totally null and geodesic 2-surfaces called the null strings or twistor surfaces. Geometric properties of these null strings are discussed. It is shown, that spaces considered are hyperheavenly spaces (HH-spaces) or, if one of the parts of the Weyl tensor vanishes, heavenly spaces (H-spaces). The explicit complex metrics admitting null Killing vectors are found. Some Lorentzian and ultrahyperbolic slices of these metrics are discussed.


Introduction
Complex methods in general theory of relativity have attracted a great interest for many years. Null tetrad formalism, twistor analysis and finally heavenly and hyperheavenly spaces (H-spaces and HH-spaces) play an important role in physics and mathematics. Presented work uses the heavenly and hyperheavenly formalism in analysis of the metrics admitting the null Killing vector.
Hyperheavenly spaces (HH-spaces) was introduced in 1976 in famous work [1] by J.F. Plebański and I. Robinson as a natural generalization of the heavenly spaces (Hspaces). Hyperheavenly spaces with cosmological constant Λ are complex spacetimes with algebraically degenerate self-dual or anti-self-dual part of the Weyl tensor satisfying the vacuum Einstein equations with cosmological constant. The transparent adventage of hyperheavenly spaces theory is the reduction of Einstein equations to one, nonlinear differential equation of the second order, i.e. hyperheavenly equation. It seemed, that finding new real vacuum solutions of Einstein field equations with the signature (+++−) was only a matter of time. It was enough to solve the hyperheavenly equations and then to find Lorentzian slices of respective complex spacetimes. This reaserch programme has been named Plebański programme. Unfortunatelly, obtaining the real slices appeared to be more difficult then anyone has ever suspected. In order to understand better the problem, the structure of hyperheavenly spaces together with their spinorial description has been investigated by J.F. Plebański, J.D. Finley III and co-workers [2] - [4]. Believing that symmetry of the spacetime simplifies the problem, Killing symmetries in heavenly and hyperheavenly spaces have been considered [5] - [8]. Lorentzian slices remained elusive, except some examples [4], [9] - [13] and discussions [14] no general techniques have been presented. Probably it was the reason why hyperheavenly machinery became less popular in nineties.
Within five last years hyperheavenly spaces found their place in deep mathematical considerations. Their connection to Walker and Osserman geometry has been noticed in 2008. A few transparent results have been obtained with help of hyperheavenly formalism [15,16]. It appeared, that HH-spaces are the most natural tool in investigating real space of the neutral (ultrahyperbolic) signature (+ + −−). Moreover, a few works devoted to Killing symmetries in heavenly and hyperheavenly spaces appeared [17] - [20]. These papers generalized the previous ideas of J.F. Plebański, J.D. Finley III, S. Hacyan and S.A. Sonnleitner. Between Killing vectors especially useful are these ones, which are tangent to self-dual null string. The existence of such (null) Killing vectors simplyfies the hyperheavenly equation, making it solvable in majority of cases.
The main aim of our work is to find all complex hyperheavenly and heavenly metrics admitting null homothetic and isometric Killing symmetry. Such metrics appear to be important in (+ + −−) real geometries. However, the existence of a null Killing vector appeared to be helpfull for finding the Lorentzian slices [18]. We want to develop this idea and examine all possible Lorentzian slices of the complex spacetimes admitting null Killing vector.
It is well known [14] that if a complex spacetime admits any real Lorentzian slice then both self-dual and anti-self-dual parts must be of the same Petrov -Penrose type. So if this complex spacetime is a hyperheavenly space (with or without Λ) then by the Goldberg -Sachs theorem it admits both self-dual and anti-self-dual congruences of null strings which intersect each other and these intersections constitute the congruence of null geodesics. To assume the existence of null Killing vector field and identify this field with congruence of null geodesics seems to be the natural first step in investigating Lorentzian slices.
Our paper is organized as follows.
In section 2 we investigate the general properties of Killing vectors, especially of null ones. Some useful theorems are given and connection between null Killing vector and null strings is pointed out. Then the detailed discussion on the possible Petrov -Penrose types admitting null Killing symmetry is presented. Section 3 is a concise summary of the properties of hyperheavenly spaces. The main goal of our work is to present explicit form of the metrics with null Killing symmetry. The results are gathered in sections 4 and 5. There are seven different hyperheavenly metrics with null isometric or homothetic Killing vector and five different heavenly metrics. In section 6 we discuss the possible real slices of the metrics found in preceding sections. Some two-sided Walker and globally Osserman spaces are obtained. The Lorentzian slices of the type [II] and [D] are found. Concluding remarks ends our paper.
2 Null Killing vectors and null strings.
2.1 Killing equations and their integrability conditions in spinorial formalism.
The system of Killing equations are given by The Killing vector is said to be conformal, if χ = const, homothetic if χ = χ 0 = const = 0 and isometric if χ = 0. For our purposes it is useful to present the Killing equations and their integrability conditions in the spinorial formalism. Let (e 1 , e 2 , e 3 , e 4 ) be a null tetrad and (∂ 1 , ∂ 2 , ∂ 3 , ∂ 4 ) its inverse basis. Then the respective spinorial images are given by We use the following rules of manipulating the spinorial indices where ∈ AB and ∈ȦḂ are the spinor Levi-Civita symbols Correspondence between the null tetrad formalism and spinorial formalism is realized with the use of the spin-tensor g aAḂ which is defined by the relation g AḂ = g AḂ a e a . It is easy to see that − 1 2 g aAḂ g bAḂ = δ a b and − 1 2 g aAḂ g aCḊ = δ A C δḂ D . The operators ∂ AḂ and ∇ AḂ are the spinorial images of operators ∂ a and ∇ a , respectively, given by A conformal Killing vector K can be written as Components K a and K AḂ are related by Conformal Killing equations with conformal factor χ in spinorial form read which is equivalent to the following system of equations From (2.9a) and (2.9b) it follows that The integrability conditions of (2.9a) and (2.9b) in Einstein space (C ABĊḊ = 0, R = −4Λ) consist of the following equations

Null strings via null Killing vectors.
The existence of a null Killing vector has a significant influence on the geometry of the space. To explain this we first note that the null Killing vector can be presented in the form where µ A and νḂ are some nonzero spinors. Moreover, it is well known that every spinor symmetric in all indices can be decomposed according to the formula where Ψ

(i)
A are some basic spinors. In particular, there exist spinors A A , B A , AȦ and BȦ such that We prove the following Lemma 2.1 Spinors l AB and lȦḂ can be brought to the form l AB = µ (A B B) and lȦḂ = ν (Ȧ BḂ ) without any loss of generality.

Theorem 2.2
Let the null Killing vector K AḂ be of the form (2.13). Then the two-dimensional self-dual holomorphic distribution {µ A νḂ, µ A ρḂ}, νḂρḂ = 0, is integrable and its integral manifolds constitute the congruence of self-dual null strings and the anti-self-dual distribution {µ A νḂ, σ A νḂ}, µ A σ A = 0, is also integrable and its integral manifolds constitute the congruence of anti-self-dual null strings. Moreover, both Weyl spinors C ABCD and CȦḂĊḊ are algebraically special with µ A and νḂ being the undotted and dotted, respectively, multiple Penrose spinors.
Proof Contracting (2.16) with µ A µ C and remembering that A A = µ A we get This means that the spinor µ A defines a congruence of self-dual null strings in the sense that the 2-dimensional holomorphic distribution {µ A νḂ, µ A ρḂ}, νȦρȦ = 0 is integrable and its integrable manifolds constitute the congruence of self-dual null strings. From the complex Sachs-Goldberg theorem it follows, that C ABCD is algebraically special and µ A is multiple Penrose dotted spinor, i.e.
Analogously we prove that In particular from Theorem 2.2 it follows that the integral curves of a null Killing vector are given by the intersection of self-dual and anti-self-dual congruences of null strings. Note that where Θ A and ΘȦ describe the optic properties of the anti-self-dual and self-dual null strings, respectively. Indeed, if ΘȦ = 0 then the self-dual null strings are parallelypropagated, if Θ A = 0 then anti-self-dual null strings are parallely propagated. Inserting (2.21), A A = µ A , AȦ = νȦ into (2.16), after some straightforward calculations we obtain Let us prove another important theorem.

Theorem 2.3
Assume, that at least one of the C ABCD or CȦḂĊḊ is nonzero. Then Assume, that C ABCD = 0. Then from (2.12f) it follows, that that ∇Ȧ A χ is the quadruple Debever-Penrose spinor. However, as is well known, two quadruple DP spinors are necessarily lineary dependent so ∇ A1 χ has to be proportional to ∇ A2 χ or, equivalently Acting on (2.23) with ∇Ḃ B and using (2.12e) one quickly obtains Hence if Λ = 0 then ∇Ḃ B χ = 0. Finally, using (2.12e)) we get χ = 0 what proves (i). If Λ = 0, then still ∇Ȧ N χ is a quadruple DP-spinor. However, we proved that µ N is a multiple DP-spinor, so it must be with some χȦ. Inserting (2.25) into (2.12e) and contracting with µ B we arrive at the conclusion ΘȦχḂ = 0, so if we want to maintain possible conformal symmetries, the selfdual null string defined by the (conformal) Killing vector must be nonexpanding, ΘȦ = 0. Consequently lȦḂ = 0. Inserting this into (2.12b) and contracting it with ∈ṘṠ we finally get χȦ = 0. From (2.25) it follows that ∇ NȦ χ = 0 and this proves (ii).
Summing up, null conformal symmetries can appear only in the Einstein spaces with C ABCD = 0 = CȦḂĊḊ, i.e. in the de-Sitter space (with Λ = 0) or in Minkowski space (with Λ = 0). We do not consider these spaces here. The null Killing vector field defines congruence of null (complex) geodesics. The optical properties of Killing vector field can be easily obtained. One gets expansion := 1 2 ∇ a K a = 2χ 0 (2.26a) Thus we conclude, that null homothetic Killing field defines null geodesic congruence with nonzero expansion, twist and shear, while null isometric Killing field is nonexpanding, nontwisting and shearfree. Gathering above considerations: we reduced the problem of null Killing vectors in Einstein space to the set of equations form of the Killing vector : Killing equation: Algebraic degeneration conditions C ABCD µ A µ B µ C = 0 and CȦḂĊḊνȦνḂνĊ = 0 can be combined with (2.12a) and (2.12b). After some work we obtain where Σ,Σ, Ω andΩ are defined by the relations We end this subsection by pointing two relations, essential in further analysis. Contracting (2.29a) with µ S µ T and using (2.27d) we obtain Analogously, contracting (2.29b) with νṠνṪ then using (2.27e) we obtain Now we are ready to discuss the possible algebraic types admitting null Killing vector.

Null homothetic symmetries.
Here we assume χ 0 = 0, what immediately gives Λ = 0. Simple analysis of equations (2.30) -(2.31) together with (2.27f) bring us to the conlcusion, that the only possibilities are • ΘȦ = 0 (self-dual null string is nonexpanding), µ A Θ A = 0 (anti-self-dual null string is necesarilly expanding, more even, expansion Θ A cannot be proportional to DPspinor µ A ) • Θ A = 0 (anti-self-dual null string is nonexpanding), νȦΘȦ = 0 (self-dual null string is necesarilly expanding, more even, expansion ΘȦ cannot be proportional to DPspinor νȦ) Of course, both possibilities constitute Eintein spaces with the same geometric properties. It is enough to consider only one of them with detailes, say ΘȦ = 0. From (2.29b) we conclude, thatΣ =Ω = 0. Careful analysis of (2.29a) gives Ω = 0. From (2.28a) and (2.28b) we obtain Contracting (2.33) with Θ T we obtain Θ T ∇ RȦ Θ T = 0, so Θ T defines the congruence of the self-dual null strings. But we proved earlier (see Theorem 2.2), that self-dual null string is defined by µ T . The number of independent congruences of self-dual null strings is equal the number of multiple undotted DP-spinors, so there are infinitely many independent congruences of self-dual null strings in the heavenly spaces, two in the self-dual type [D] and only one in the self-dual types [II, III, N]. But here we examine self-dual type [N], so there is only one congruence of the null-strings. It means, that Θ T must be proportional to µ T or Θ T µ T = 0 → χ 0 = 0. This contradicts our assumption, that χ 0 = 0. It proves, that the only possible Petrov types which admitt null homothetic symmetries are [III, −] ⊗ [N, −]. Self-dual null string is nonexpanding, anti-self-dual null string must be expanding.
[Remark: considering the second possibility with Θ A = 0 we obtain possible Petrov types [N, −] ⊗ [III, −], but still type [III] corresponds to nonexpanding null string, and the type [N] corresponds to expanding null string].
All possible types via geometric properties of the null strings are presented in the  table below: self-dual null string is self-dual null string is All independent possibilities are given in detailes in the subsections:
All possible types are gathered in the table below: self-dual null string is self-dual null string is  3 Hyperheavenly spaces.
The considerations from the previous section allow to establish all possible algebraic types of the spaces, which admit the null Killing symmetry. Main aim of present paper is to present the explicit metrics with such symmetries. Due to Theorem 2.2, null Killing vector defines the congruence of both self-dual and anti-self-dual null strings and extorts the algebraic degeneration of both self-dual and anti-self-dual part of the Weyl curvature spinor. Let us remind the definition of hyperheavenly space.

Definition 3.1
Hyperheavenly space (HH-space) with cosmological constant is a 4 -dimensional complex analytic differential manifold endowed with a holomorphic Riemannian metric ds 2 satisfying the vacuum Einstein equations with cosmological constant and such that the selfdual or anti -self -dual part of the Weyl tensor is algebraically degenerate. These kind of spaces admits a congruence of totally null, self-dual (or anti-self-dual, respectively) surfaces.
In general, hyperheavenly spaces require only self-dual (or anti-self-dual) congruences of null strings. The spaces which admitt the null Killing vector are equipped in both self-dual and anti-self-dual congruences of null strings, so they are automaticaly hyperheavenly spaces. In hyperheavenly spaces, vacuum Einstein equations can be reduced to one, nonlinear, partial differential equation of the second order, for one holomorphic function. This equation is called hyperheavenly equation. It seems, that using the hyperheavenly formalism in order to obtain the explicite metrics which admitt the null Killing vector is most natural.
The existence of the null strings allows to introduce some useful tetrad and the coordinate system. The self-dual null string generated by the null Killing vector is given by the equation µ B µ C ∇Ȧ B µ C = 0 which is equivalent to the Pfaff system µ A g AḂ = 0. Choosing the spinorial basis in such a manner, that µ A = (0, µ 2 ), µ 2 = 0 we arrive to conclusion, that the self-dual null string is defined by the Pfaff system e 1 = 0 , e 3 = 0 ⇔ g 2Ȧ = 0 (3.1) (the surface element of the null string is given by e 1 ∧ e 3 ). A null tetrad (e 1 , e 2 , e 3 , e 4 ) and a coordinate system (qȦ, pḂ) can be always chosen so that where φ and QȦḂ = QḂȦ are holomorphic functions. Coordinates qȦ label the null strings, hence pȦ are coordinates on them. Dual basis is given by Of course, for consistency with (2.3), the rules to raise and lower spinor indices in spinorial differential operators read ∂Ȧ = ∂Ḃ ∈ȦḂ, ∂Ȧ =∈ḂȦ ∂Ḃ, ðȦ = ðḂ ∈ȦḂ, ðȦ =∈ḂȦ ðḂ, so The metric ds 2 is given by The congruence of null strings have some invariant properties. Investigating the equation ∇Ȧ B µ C = ZȦ B µ C + ∈ BC ΘȦ with µ 1 = 0, µ 2 = 0 we easily find, that If ΘȦ = 0 ⇐⇒ Γ 112Ȧ = 0 then self-dual null strings are parallely propagated. The hyperheavenly spaces based on such null strings are called nonexpanding. If the null string is not parallely propagated (ΘȦ = 0 ⇐⇒ Γ 112Ȧ = 0), the corresponding hyperheavenly space is called expanding. Vacuum Einstein equations impose some constraints on φ and QȦḂ. The final forms of the φ and QȦḂ are esentially different in expanding and nonexpanding hyperheavenly spaces.

Nonexpanding hyperheavenly spaces
The reduction of Einstein equations brings us to Einstein equations can be reduced to nonexpanding hyperheavenly equation with Λ where FȦ, NȦ and γ are arbitrary functions of qĊ only (constant on each null string), Λ is a cosmological constant and Θ = Θ(pȦ, qḂ) is the key function. The metric is given by (3.6) with φ and QȦḂ in the form (3.9).
Using the connection forms, calculated explicitly in [19] one can note, that in expanding hyperheavenly spaces expansion of the congruence of the self-dual null strings is proportional to nonzero spinor JȦ, namely ΘȦ = − √ 2φ −1 µ 2 JȦ. Hence ΘȦ = 0.  [19]. Killing vector has the form and The key function and the curvature reads where F = F (x, w, t) and γ = γ(w, t) are arbitrary functions of their arguments, such that F xxx = 0 and γ t = 0. Inserting the key function W (4.3) into hyperheavenly equation we get where w = τ w, t = τ t oraz γ = τ −2 γ. The general solution of the equation (4.5) is not known. The metric has the form (remember, that η = xφ) Equation (4.5) under the additional assumption F xxx = 0 can be easily solved. Using gauge freedom, which is still available (see [19] for detailes) one gets where f = f (w, t) is an arbitrary function. The metric is

Heavenly spaces of the type [III] n ⊗ [−] e
The second possible heavenly reduction of the hyperheavenly space of the type [N] e ⊗ [III] n with null homothetic symmetry is the heavenly space of the type [−] e ⊗ [III] n . Formally it is enough to set C (1) = 0 ⇐⇒ γ t = 0 in subsection 4.1. However, equation (4.5) still is hard to solve. It appeared to be much more convenient to attack the problem from the opposite side and consider the space of the type [III] n ⊗ [−] e . As a starting point we take the nonexpanding hyperheavenly spaces and we set CȦḂĊḊ = 0. It allows us to take the general key function as a third-order polynomial in pȦ coordinates (see [17,18] for detailes). The metric appeared to be two-sided Walker [15]. Using general results from [18] and [15] (especially Theorem 5.1 from [15]) we find the form of the Killing vector K = 2χ 0 pȦ ∂ ∂pȦ (4.10) and the spinors l AB and lȦḂ The key function and the curvature where X = X(qṀ ) and Y = Y (qṀ ) are arbitrary functions. The metric [Note, that the conditions CȦḂĊḊ = 0 = Λ assure the infinitely many congruences of nonexpanding anti-self-dual null strings, that is why the space considered belongs to the two-sided Walker class. However, the anti-self-dual null string generated by the null Killing vector is expanding.] 5 Metrics admitting null isometric symmetries. 5 [8] (with Λ = 0) and then in [19] (with Λ = 0). The Killing vector reads The key function and the curvature are The metric reads After inserting the key function (5.3) into the hyperheavenly equation we get In order to maintain the type [II] ⊗ [II] we have to assume W t = 0 and W φφφφ = 0. The general solution of the equation (5.6) is not known. Its reduction to canonical form is realized by the transformation Considering the key function W as a function of the variables (z, s, w) we obtain the equation Multiplying the equation (5.8) by φ φ 3 + Λ 6µ 0 −2 one can bring it to the form From (5.9) we infer the existence of the potential Σ = Σ(z, s, w) such that From the (5.10b) one can calculate W Inserting it into (5.10a) we arrive to the equation Of course, φ has to be considered as a function of coordintates s and z, φ = φ(z, s), according to (5.7). The integral in (5.7) can be calculated in all subcases, leading to the condition z = z(s, φ). Unfortunatelly, the inverse function φ = φ(z, s) is an elementary function only if Λ = 0. However, if Λ = 0 more efficient transformation can be proposed.

Equation (5.17) is equivalent to the Euler-Poisson-Darboux equation (EPD equation) and
its solutions have been discussed in literature [21]. The form of the metric (5.14) and the equation (5.17) are especially useful in obtaining the Lorentzian slices (see section 6).

Spaces of the type [D] e ⊗ [D] e
From the previous subsection one can easily obtain the general metric for the type [D] e ⊗ [D] e with Λ. Self-dual type [D] we get after setting C (1) = 0 ⇒ W t = 0 ⇒ W = W (φ, w). General solution of the hyperheavenly equation (5.6) reads W = µ 0 f 1 (w)φ 3 + f 2 (w)φ − (Λ/3)f 1 (w) and it automatically causes anti-self-dual type beeing of the type [D]. Moreover, it can be proved that arbitrary functions f 1 and f 2 can be gauged to zero without any loss of generality. [We do not prove that fact here, but it can be easily done by using the results from [19] admitts, together with null isometric Killing vector ∂ η , three other isometric Killing vectors ∂ w , ∂ t and w∂ w − η∂ η and -if Λ = 0 -one homothetic Killing vector 2 3 χ 0 (2t∂ t − φ∂ φ + η∂ η ).
If Λ = 0 the metric (5.18) can be easily transformed to coordinate system (x, y, u, v) defined by (5.13). According to (5.15) function M = 0 and the metric reads and

Spaces of the type
The key function and the curvature where f = f (z, w) and g = g(w, t) are arbitrary functions of their variables. The metric In all formulas Λ = 0. Particular types are characterized by • type [III] e ⊗ [III] e : α 0 = 0, α 0 can be re-gauged to 1 without any loss of generality In this case we deal with the hyperheavenly types [III,N] n ⊗ [N] e with nonexpanding self-dual null string defined by the null Killing vector. Anti-self-dual null string is still expanding. These metrics have been discussed in [18]. The Killing vector has the form The key function and the curvature (1) δ˙2Ȧδ˙2Ḃδ˙2Ċδ˙2Ḋ ,Ċ (1) = 2 p˙2 S(q˙2, q˙1p˙2p˙2) p2p2p2p2 (5.27) where N = N (qṀ ) and S = S(q˙2, q˙1p˙2p˙2) are arbitrary functions, and F 0 is constant. The metric The cosmological constant must be necesarilly zero here Λ = 0, hyperheavenly metrics are characterized by The heavenly reductions of the metric obtained above are especially interesting here, because they offer two different null string geometries. One can get the metrics of the type [III,N] n ⊗ [−] e in which types [III,N] corresponds to the nonexpanding self-dual null string and conformally flat, anti-self-dual part correspond to the expanding anti-self-dual null string. The second possible heavenly reduction [−] n ⊗ [N] e has different geometry of null strings. In this case expanding null string (anti-self-dual one) corresponds to the type [N] and nonexpanding null string (self-dual one) corresponds to the conformally flat part.
In order to obtain the heavenly metrics of the types [III,N] n ⊗ [−] e , one must set S(q˙2, q˙1p˙2p˙2) = f (q˙2) q˙1p˙2p˙2 where f is an arbitrary function of the variable q˙2. (It seems, that this is to strong condition and it is enough to set (p˙2 S) p2p2p2p2 = 0, but there is unused gauge freedom, which allows to simplify the function S). Heavenly metrics of the type [−] n ⊗ [N] e can be obtained by setting F 0 = 0 = N (once again condition N = 0 is stronger then necessary C (1) = 0, but N = 0 can be obtained by using gauge freedom). Finally The key function and the curvature where f = f (w, t) is an arbitrary function and the metric reads Of course, the metric (5.33) is equivalent to the metric (5.28) with F 0 = 0 = N .

Spaces of the type
The last case is characterized by both self-dual and anti-self-dual null strings being nonexpanding. The only possible types are [N, −] n ⊗ [N, −] n . These metrics have been found in [18]. The Killing vector takes the form with spinors l AB and lȦḂ l AB = 0 , lȦḂ = 0 (5.35) The key function and the curvature Θ = 1 2 N (qṀ ) p˙2p˙2 + A(p˙2, q˙2) (5.36) where N = N (qṀ ) and A = A(p˙2, q˙2) are arbitrary functions. The metric reads 6 Real slices. 6.1 Real slices with neutral signature (+ + −−) All metrics presented in sections 4 and 5 are holomorphic. It is an easy matter to carry over all the results to the case of real spaces of the signature (+ + −−). Instead of the holomorphic objects (spinors, null strings, tetrads, coordinates, etc.) we deal with the real smooth objects. A real spaces with the neutral signature play an important role in Walker and Osserman geometry. A few papers dealing with such geometries appeared recently [22] - [24]. However, it was hyperheavenly formalism which allowed to obtain transparent results in Walker and Osserman geometries. For example, a new class of metrics admitting self-dual and anti-self-dual, parallely propagated null strings (two-sided Walker spaces) has been found in [15]. These spaces have a natural generalization: if only one of the null strings is parallely propagated, we deal with so called sesqui-Walker spaces. Such spaces have been defined and investigated in [24]. Probably the most distinguished success of the hyperheavenly methods in Osserman geometry was finding all algebraically degenerate metrics of the globally Osserman space, which do not have the Walker property, i.e. they do not admit any parallely propagated null strings [16].
Some of the metrics presented in sections 4  real, this step changes automatically the signature of the metric making (5.14) Lorentzian. The second reason is that the metric (5.14) does not depend directly on the key function W , but on the function M . The relation between this two functions is given by (5.15) and it contains the imaginary unit. However, by twice differentiation the hyperheavenly equation ( It is worth to note, that this construction succeeds only in the vacuum case. If cosmological constant Λ = 0 we have not been able to find the Lorentzian slice.
No spaces with null homothetic symmetries admit Lorentzian slices.
In the presented paper the null Killing vectors (isometric and homothetic) in complex spacetime have been considered. The connection between null Killing vectors and null strings has been pointed in section 2. Because of the existence of the null strings the most natural apparatus in investigating null Killing vectors appeared to be hyperheavenly and heavenly spaces. After short summary of the structure of hyperheavenly spaces (section 3), we have been able to present all possible metrics admitting null Killing vector. Only two of them with Λ = 0 has been reduced to the equation (5.12), but this reduction has obvious disadvantages. Like in the previous case, we will deal with this equation soon. The transparent results are the metrics (4.9) and (4.14) which constitute all heavens with null homothetic symmetry. These cases have been considered in [26], but without giving any explicit form of the metric. We were able to integrate the problem completely.
Probably the most interesting from the physical point of view is another example of real Lorentzian slice of the complex metric. The first such an example has been presented in [18]. Here we have been able to find the Lorentzian slices of the types [II] e ⊗ [II] e and [D] e ⊗[D] e with Λ = 0. They are given by the metric (5.14) which depends on one function M of three variables satisfying the equation (5.17). Such a metric has been presented earlier (see [27], or in a concise form [25]).
Both these examples gave some valuable notes about obtaining Lorentzian slices. In the first of them the condition CȦḂĊḊ =C ABCD has been succesfully used, in the second one a resonable using of imaginary unit appears to be essential. Nonetheless, the case with nonzero cosmological constant is still unsolved. Taking into considerations the optical properties of the congruneces of null geodesics defined by the null isometric Killing vector (2.26a) -(2.26c) we conclude that all such slices must belong to the Kundt class ( [25], xxxi). The vacuum Einstein field equations with cosmological constant for the Kundt class have been gathered in [25]  We hope, that further investigation of the structure of complex spacetime allows to find some effective and more general techniques of obtaining real Lorentzian slices.