Partner Symmetries, Group Foliation and ASD Ricci-Flat Metrics without Killing Vectors

We demonstrate how a combination of our recently developed methods of partner symmetries, symmetry reduction in group parameters and a new version of the group foliation method can produce noninvariant solutions of complex Monge-Ampère equation (CMA) and provide a lift from invariant solutions of CM...

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Дата:	2013
Автори:	Sheftel, M.B., Malykh, A.A.
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Опубліковано:	Інститут математики НАН України 2013
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
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Цитувати:	Partner Symmetries, Group Foliation and ASD Ricci-Flat Metrics without Killing Vectors / M.B. Sheftel, A.A. Malykh // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 28 назв. — англ.

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spelling	irk-123456789-1493672019-02-22T01:23:30Z Partner Symmetries, Group Foliation and ASD Ricci-Flat Metrics without Killing Vectors Sheftel, M.B. Malykh, A.A. We demonstrate how a combination of our recently developed methods of partner symmetries, symmetry reduction in group parameters and a new version of the group foliation method can produce noninvariant solutions of complex Monge-Ampère equation (CMA) and provide a lift from invariant solutions of CMA satisfying Boyer-Finley equation to non-invariant ones. Applying these methods, we obtain a new noninvariant solution of CMA and the corresponding Ricci-flat anti-self-dual Einstein-Kähler metric with Euclidean signature without Killing vectors, together with Riemannian curvature two-forms. There are no singularities of the metric and curvature in a bounded domain if we avoid very special choices of arbitrary functions of a single variable in our solution. This metric does not describe gravitational instantons because the curvature is not concentrated in a bounded domain. 2013 Article Partner Symmetries, Group Foliation and ASD Ricci-Flat Metrics without Killing Vectors / M.B. Sheftel, A.A. Malykh // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 28 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 35Q75; 83C15 DOI: http://dx.doi.org/10.3842/SIGMA.2013.075 http://dspace.nbuv.gov.ua/handle/123456789/149367 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We demonstrate how a combination of our recently developed methods of partner symmetries, symmetry reduction in group parameters and a new version of the group foliation method can produce noninvariant solutions of complex Monge-Ampère equation (CMA) and provide a lift from invariant solutions of CMA satisfying Boyer-Finley equation to non-invariant ones. Applying these methods, we obtain a new noninvariant solution of CMA and the corresponding Ricci-flat anti-self-dual Einstein-Kähler metric with Euclidean signature without Killing vectors, together with Riemannian curvature two-forms. There are no singularities of the metric and curvature in a bounded domain if we avoid very special choices of arbitrary functions of a single variable in our solution. This metric does not describe gravitational instantons because the curvature is not concentrated in a bounded domain.
format	Article
author	Sheftel, M.B. Malykh, A.A.
spellingShingle	Sheftel, M.B. Malykh, A.A. Partner Symmetries, Group Foliation and ASD Ricci-Flat Metrics without Killing Vectors Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Sheftel, M.B. Malykh, A.A.
author_sort	Sheftel, M.B.
title	Partner Symmetries, Group Foliation and ASD Ricci-Flat Metrics without Killing Vectors
title_short	Partner Symmetries, Group Foliation and ASD Ricci-Flat Metrics without Killing Vectors
title_full	Partner Symmetries, Group Foliation and ASD Ricci-Flat Metrics without Killing Vectors
title_fullStr	Partner Symmetries, Group Foliation and ASD Ricci-Flat Metrics without Killing Vectors
title_full_unstemmed	Partner Symmetries, Group Foliation and ASD Ricci-Flat Metrics without Killing Vectors
title_sort	partner symmetries, group foliation and asd ricci-flat metrics without killing vectors
publisher	Інститут математики НАН України
publishDate	2013
url	http://dspace.nbuv.gov.ua/handle/123456789/149367
citation_txt	Partner Symmetries, Group Foliation and ASD Ricci-Flat Metrics without Killing Vectors / M.B. Sheftel, A.A. Malykh // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 28 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT sheftelmb partnersymmetriesgroupfoliationandasdricciflatmetricswithoutkillingvectors AT malykhaa partnersymmetriesgroupfoliationandasdricciflatmetricswithoutkillingvectors
first_indexed	2025-07-12T21:57:38Z
last_indexed	2025-07-12T21:57:38Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 9 (2013), 075, 21 pages Partner Symmetries, Group Foliation and ASD Ricci-Flat Metrics without Killing Vectors Mikhail B. SHEFTEL † and Andrei A. MALYKH ‡ † Department of Physics, Boğaziçi University 34342 Bebek, Istanbul, Turkey E-mail: mikhail.sheftel@boun.edu.tr URL: http://www.phys.boun.edu.tr/faculty_wp/mikhail_sheftel.html ‡ Department of Numerical Modelling, Russian State Hydrometeorlogical University, 98 Malookhtinsky Ave., 195196 St. Petersburg, Russia E-mail: andrei-malykh@mail.ru Received June 14, 2013, in final form November 19, 2013; Published online November 27, 2013 http://dx.doi.org/10.3842/SIGMA.2013.075 Abstract. We demonstrate how a combination of our recently developed methods of part- ner symmetries, symmetry reduction in group parameters and a new version of the group foliation method can produce noninvariant solutions of complex Monge–Ampère equation (CMA) and provide a lift from invariant solutions of CMA satisfying Boyer–Finley equation to non-invariant ones. Applying these methods, we obtain a new noninvariant solution of CMA and the corresponding Ricci-flat anti-self-dual Einstein–Kähler metric with Euclidean signature without Killing vectors, together with Riemannian curvature two-forms. There are no singularities of the metric and curvature in a bounded domain if we avoid very spe- cial choices of arbitrary functions of a single variable in our solution. This metric does not describe gravitational instantons because the curvature is not concentrated in a bounded domain. Key words: Monge–Ampère equation; Boyer–Finley equation; partner symmetries; sym- metry reduction; non-invariant solutions; group foliation; anti-self-dual gravity; Ricci-flat metric 2010 Mathematics Subject Classification: 35Q75; 83C15 1 Introduction In his pioneer paper [21], Plebañski demonstrated that anti-self-dual (ASD) Ricci-flat metrics on four-dimensional complex manifolds are completely determined by a single scalar potential which satisfies his first or second heavenly equation. Such metrics are solutions to complex vacuum Einstein equations. Real four-dimensional Kähler ASD metrics ds2 = 2 ( u11̄dz 1dz̄1 + u12̄dz 1dz̄2 + u21̄dz 2dz̄1 + u22̄dz 2dz̄2 ) (1.1) that solve the vacuum Einstein equations with Euclidean (Riemannian) signature are governed by a scalar real-valued potential u = u ( z1, z2, z̄1, z̄2 ) which satisfies complex Monge–Ampère equation (CMA) u11̄u22̄ − u12̄u21̄ = 1. (1.2) A modern justification for this conclusion one can find in the books by Mason and Wood- house [16] and Dunajski [4]. Here and henceforth, subscripts denote partial derivatives with respect to corresponding variables whereas the bar means complex conjugation, e.g. u11̄ = ∂2u/∂z1∂z̄1 and suchlike. The only exception is Section 3 where subscripts of generators X are used to designate different vector fields. mailto:mikhail.sheftel@boun.edu.tr http://www.phys.boun.edu.tr/faculty_wp/mikhail_sheftel.html mailto:andrei-malykh@mail.ru http://dx.doi.org/10.3842/SIGMA.2013.075 2 M.B. Sheftel and A.A. Malykh Among ASD Ricci-flat metrics, the most interesting ones are those that describe gravitational instantons which asymptotically look like a flat space, so that their curvature is concentrated in a finite region of a Riemannian space-time (see [4] and references therein). The most important gravitational instanton is K3 which geometrically is Kummer surface [1], for which an explicit form of the metric is still unknown while many its properties and existence had been discovered and analyzed [8, 28]. A characteristic feature of the K3 instanton is that it does not admit any Killing vectors, that is, no continuous symmetries which implies that the metric potential should be a noninvariant solution of CMA equation. As opposed to the case of invariant solutions, for noninvariant solutions of CMA there should be no symmetry reduction in the number of inde- pendent variables. Since standard methods of Lie group analysis of PDEs provide only invariant solutions, which implies symmetry reduction in the solutions and hence in the metric (1.1), they cannot be applied for obtaining noninvariant solutions to the CMA equation and ASD Ricci-flat metrics without Killing vectors. Thus, to obtain at least some pieces of K3 metric explicitly one needs a technique for deriving non-invariant solutions of multi-dimensional non-linear equations. In the previous papers [10, 11, 13] we have developed methods for obtaining noninvariant solutions though still remaining in the symmetry framework. We extended the Lax equations of Mason and Newman for CMA [15, 16] by supplementing them with another pair of linear equations, so that CMA becomes an algebraic consequence of these equations, whereas the original Lax pair generated only differential consequences of CMA with no distinction between elliptic, hyperbolic and homogeneous versions of CMA. The equation, determining symmetry characteristics [19] for CMA, now appears as the integrability condition of our linear equations. Moreover, the symmetry condition has a two-dimensional divergence form and therefore uniquely determines locally a potential function which turns out to be a solution to the symmetry con- dition, that is, the potential of a symmetry is also a symmetry, which is a characteristic feature of a more general class of Monge–Ampère equations [25]. We called this pair of symmetries, the original one and the corresponding potential, partner symmetries and applied them to generate noninvariant solutions of CMA and corresponding heavenly metrics without Killing vectors. We discovered that the equations connecting partner symmetries can be treated as an invariance condition for solutions of CMA with respect to a certain nonlocal symmetry constructed from partner symmetries and nonlocal recursion operators [10]. This was close but not identical to the idea of hidden symmetries by Dunajski and Mason [6]. The invariance of solutions under the nonlocal symmetry, unlike the invariance under a local symmetry, does not imply the symmetry reduction in the number of independent variables, so such solutions are noninvariant in the usual sense. The problem of finding solutions of CMA by solving the equations for partner symmetries was facilitated by the observation that the full system of such equations provides a lift from invariant solutions of CMA to its noninvariant solutions [12, 13, 24]. As is shown in [2], any three dimensional reduction of CMA leads to either translationally invariant solutions satisfying 3-dimensional Laplace equation or to rotationally invariant solutions satisfying elliptic Boyer– Finley (BF) equation. Earlier, in [13], we showed that there exists a lift of translationally invariant solutions of CMA to noninvariant solutions. In [12] we obtained a lift from hyperbolic version of Boyer–Finley equation to noninvariant solutions of the hyperbolic CMA, with 1 re- placed by −1 on the right-hand side of (1.2). In this paper we will show that there also exist rotationally invariant solutions, related to solutions of elliptic Boyer–Finley equation, which can be lifted to noninvariant solutions of the elliptic CMA (1.2). Therefore, one may start with a simpler problem of finding translationally or rotationally invariant solutions and then, using the partner symmetries, lift them to noninvariant solutions. Another modification of the method of partner symmetries was, by using Lie equations, to introduce explicitly symmetry group parameters (for commuting symmetries) as additional independent variables of the problem instead of symmetry characteristics [13]. This trick allowed Partner Symmetries, Group Foliation and ASD Ricci-Flat Metrics 3 us to discover an integrability condition of the equations for partner symmetries. Furthermore, this made possible using symmetry reductions in group parameters without symmetry reductions in “physical variables” in order to simplify the equations to be solved without ending up with invariant solutions. In this paper, we apply to the full system of equations for partner symmetries of CMA another solution tool, group foliation [14, 18, 23], which we used before for CMA and the Boyer–Finley equation [2]. The idea of the group foliation belongs to S. Lie [9] and was developed by E. Ves- siot [27] and a modern presentation was given by L.V. Ovsiannikov [20]. Original differential equations are foliated with respect to a chosen symmetry (sub)group into automorphic equa- tions describing orbits of the symmetry group and resolving equations determining a collection of the orbits. The automorphic property of the first subsystem of equations means that any of its solutions can be obtained from any other solution by some transformation of the chosen symmetry subgroup. This property makes the automorphic system completely integrable if only one of its solutions can be obtained. Thus, the problem reduces to obtaining as many particular solutions of the resolving system as possible. Each solution will fix a particular automorphic system and the corresponding orbit in the solution space of original equations. Applying this method to equations of partner symmetries, we find a very large number of resolving equations for which it is extremely difficult to find even a single solution. Therefore, we use our modification of the method which utilizes to a greater degree operators of invariant differentiation. They are defined by the property that they commute with any prolongation of the symmetry generators of the Lie group chosen for the foliation and hence they map differential invariants again into differential invariants. Our discovery was that the resolving system is equivalent to the commutator algebra of operators of invariant differentiation together with its Jacobi identities, so that we can replace the problem of solving the resolving equations by the problem of solving commutator relations between the operators of invariant differentiation. An example of such approach was given in our papers [14, 18, 23]. In Section 2, we derive the extended system of six equations for partner symmetries of CMA including their integrability condition and introduce symmetry group parameters as additional independent variables. In Section 3, we determine all point symmetries of the extended system and perform its symmetry reduction with respect to two group parameters. This is not a symmetry reduction with respect to “physical” variables and so it does not imply a symmetry reduction of solutions of CMA. Our aim here is to prepare the ground for a lift of some solutions to the elliptic version of the Boyer–Finley (BF) equation to noninvariant solutions of CMA. We determine all point symmetries of the reduced system. To have BF in our system, we choose the symmetry of simultaneous rotations in complex z1- and z2-planes for a further reduction which picks up rotationally invariant solutions of CMA. Being followed by a Legendre transformation, this yields the BF equation. Meanwhile, we also keep non-reduced transformed CMA equation obtained from the integrability condition of the extended system, though in different variables involving symmetry group parameters. If we find a solution to BF which also satisfies all other equations of the extended system, this would mean a lift from rotationally invariant solutions to noninvariant solutions of CMA. Finally, we determine all point symmetries of the transformed extended system which we will need for the group foliation. In Section 4, we choose a complex conjugate pair of the symmetry generators which contain maximum number of arbitrary functions and thus generate a maximal infinite symmetry sub- group for the group foliation. We determine all first-order operators of invariant differentiation and obtain a set of all differential invariants up to the second order, inclusive. In Section 5, we perform the group foliation deriving the full set of automorphic and resolving equations. We derive also the commutator algebra of operators of invariant differentiation. There are too many resolving equations for a straightforward search of its particular solutions. 4 M.B. Sheftel and A.A. Malykh In Section 6, we find some solutions of the extended system by applying our strategy of making ansatzes, that simplify the commutator algebra of operators of invariant differentiation, rather than trying to solve directly a huge system of the resolving equations. As a by-product, we obtain new solutions of the Boyer–Finley equation. We choose more generally looking solution for lifting it to a solution of CMA. In Section 7, we apply Legendre transformations of our solution to a solution of CMA equation with a parameter-dependent right-hand side. This solution depends on three arbitrary functions of a single complex variable together with their complex conjugates. This is also a simultaneous solution of the transformed Boyer–Finley equation in somewhat different variables which is related to a symmetry reduction of CMA and hence determines its invariant solutions. Therefore, our construction provides a lift from invariant to non-invariant solutions of CMA. In Section 8, we use our solution of the CMA equation to obtain Kähler metric of Euclidean signature. This metric is anti-self-dual and Ricci-flat and have pole singularities in a bounded domain only for a special choice of arbitrary functions in our solution. By avoiding this special choice we obtain the metric without such singularities. We have also computed Riemannian curvature two-forms which do not depend on two variables and hence the curvature does not vanish outside of a bounded domain in the space of all variables. This means that our metric does not describe a gravitational instanton. Even though this simplest possible example of application of our approach does not produce an instanton metric, we believe that more refined solutions to our extended system of equations for partner symmetries and/or different chain of reductions will yield a gravitational instanton metric with no Killing vectors. In Section 9, we derive the invariance conditions for our solution under the symmetries of the parameter-dependent CMA. A detailed study of these conditions proves that generically (without special restrictions on arbitrary functions in our solution to CMA) the solution is noninvariant and hence the corresponding metric has no Killing vectors. 2 Basic equations In this section, we derive a complete set of equations for partner symmetries to provide a tool for obtaining noninvariant solutions of CMA while remaining in the symmetry group frame- work [10, 11]. Here we introduce symmetry group parameters as additional independent vari- ables in order to reserve a possibility of symmetry reduction in group parameters to simplify these equations without reduction in the number of original independent variables in CMA and thus avoiding ending up with invariant solutions. Using these additional variables also facili- tates a derivation of the integrability condition for our equations, so that the extended set of equations augmented with this integrability condition is integrable in the sense of Frobenius: all further integrability conditions are direct algebraic or differential consequences of already available equations. For the complex Monge–Ampère equation (1.2) the symmetry condition, that determines symmetry characteristics ϕ of (1.2), u11̄ϕ22̄ + u22̄ϕ11̄ − u12̄ϕ21̄ − u21̄ϕ12̄ = 0 (2.1) on solutions of CMA. Here the subscripts of ϕ denote total derivatives with respect to corre- sponding independent variables. Equation (2.1) can be set in the total divergence form (u11̄ϕ2 − u21̄ϕ1)2̄ − (u12̄ϕ2 − u22̄ϕ1)1̄ = 0. (2.2) We assume that all functions that we operate with are smooth, that is, they have continuous derivatives of any required order. Then equation (2.2) suggests local existence of potential ψ Partner Symmetries, Group Foliation and ASD Ricci-Flat Metrics 5 defined by the equations ψ1̄ = u11̄ϕ2 − u21̄ϕ1, ψ2̄ = u12̄ϕ2 − u22̄ϕ1 (2.3) in the sense that the condition (2.2) becomes just the equality of mixed derivatives (ψ1̄)2̄ = (ψ2̄)1̄, together with the complex conjugate equations ψ̄1 = u11̄ϕ̄2̄ − u12̄ϕ̄1̄, ψ̄2 = u21̄ϕ̄2̄ − u22̄ϕ̄1̄. (2.4) We do not discuss here the difficult problem of the global existence of potential ψ. A straightforward check shows that the potential ψ also satisfies symmetry condition (2.1), so that ψ is also a symmetry if ϕ is a symmetry and hence the relations (2.3) and (2.4) are recursion relations for symmetries (“partner symmetries”). Transformation (2.3) is algebraically invertible since its determinant equals one due to (1.2). Inverse transformation has the form ϕ1 = u12̄ψ1̄ − u11̄ψ2̄, ϕ2 = u22̄ψ1̄ − u21̄ψ2̄ (2.5) together with its complex conjugate ϕ̄1̄ = u21̄ψ̄1 − u11̄ψ̄2, ϕ̄2̄ = u22̄ψ̄1 − u12̄ψ̄2. (2.6) For symmetries with characteristics ϕ, ϕ̄, ψ and ψ̄, Lie equations read ϕ = uτ , ϕ̄ = uτ̄ , ψ = uσ, ψ̄ = uσ̄, (2.7) where τ , σ together with their complex conjugates τ̄ , σ̄ are group parameters. Simultaneous inclusion of several group parameters as additional independent variables implies the commuta- tivity conditions for corresponding symmetries in the form ϕτ̄ = ϕ̄τ , ψσ̄ = ψ̄σ, ϕσ = ψτ , ϕσ̄ = ψ̄τ and complex conjugates to the last two equations. We now use (2.7) to replace symmetry characteristics by derivatives of u with respect to group parameters in equations (2.3), (2.5) and their complex conjugates (2.4), (2.6) with the result uσ1̄ = u11̄uτ2 − u21̄uτ1, uσ2̄ = u12̄uτ2 − u22̄uτ1, (2.8) uτ1 = u12̄uσ1̄ − u11̄uσ2̄, uτ2 = u22̄uσ1̄ − u21̄uσ2̄, (2.9) and the complex conjugate equations uσ̄1 = u11̄uτ̄ 2̄ − u2̄1uτ̄ 1̄, uσ̄2 = u1̄2uτ̄ 2̄ − u22̄uτ̄ 1̄, (2.10) uτ̄ 1̄ = u1̄2uσ̄1 − u11̄uσ̄2, uτ̄ 2̄ = u22̄uσ̄1 − u2̄1uσ̄2. (2.11) We note that four equations (2.9) and (2.11) are algebraic consequences of other equations (2.8), (2.10) and CMA. We note also that CMA itself follows as an algebraic consequence from equa- tions (2.8), (2.10) and the first equation in (2.11). To study integrability conditions of our system, we set the first equations in (2.8) and (2.11) in the form (u11̄u2)τ = (uσ + u2uτ1)1̄, (u11̄u2)σ̄ = (u2uσ̄1 − uτ̄ )1̄. (2.12) Equations (2.12) constitute an active system since they have a second-order nontrivial integra- bility condition obtained by cross differentiation of these equations with respect to σ̄ and τ and further integration with respect to z̄1 uτ τ̄ + uσσ̄ + uσ̄2uτ1 − uσ̄1uτ2 = 0, 6 M.B. Sheftel and A.A. Malykh where the “constant” of integration can be eliminated by a redefinition of u. To make this equation self-conjugate, we multiply it with an overall factor u11̄ and then eliminate u11̄uσ̄2 and u11̄uτ2 in the last two terms of (2.12) using first equations in (2.11) and (2.8), respectively, with the final form of the integrability condition u11̄(uτ τ̄ + uσσ̄)− uτ1uτ̄ 1̄ − uσ1̄uσ̄1 = 0. (2.13) In a similar way, we obtain the alternative form of the integrability condition u22̄(uτ τ̄ + uσσ̄)− uτ2uτ̄ 2̄ − uσ2̄uσ̄2 = 0. We can choose (2.8), (2.10), (2.13) and CMA for the set of algebraically independent equations. All other equations are linearly dependent on the chosen equations. One could also check that there are no further independent second-order integrability conditions of our system of six equations. 3 Reduction of partner symmetries system for CMA Here we study point symmetries of the extended system for partner symmetries and use two symmetry reductions with respect to group parameters to simplify this system. The choice of the first reduction obeys the requirement for the reduced integrability condition (3.6) to be related to CMA (3.7) in new variables. The second symmetry reduction of the extended system results in rotational reduction of the original CMA (1.2) which yields equation (3.9) related to the Boyer–Finley equation (3.18) by Legendre transformation (3.15) combined with the differential substitution w = vt and the following integration of equations with respect to t. The transformed reduced integrability condition (3.12) becomes the Legendre transform of the Monge–Ampère equation (3.13). In this way we arrive at the system which contains both CMA and Boyer–Finley equation (BF) thus providing the possibility of a lift from rotationally invariant solutions of CMA (related to solutions of BF) to noninvariant solutions of CMA. The Legendre transformation appears as a necessary step because of the well-known relation between the rotational reduction of CMA and BF equations (see, e.g., (4.10), (4.12) in [2]). We list the generators of all point symmetries of the extended system of six equations CMA, (2.8), (2.10) and (2.13) X1 = ∂τ , X̄1 = ∂τ̄ , X2 = ∂σ, X̄2 = ∂σ̄, X3 = τ∂τ + σ∂σ, X̄3 = τ̄ ∂τ̄ + σ̄∂σ̄, X4 = z2∂2 − z̄2∂2̄ + τ̄ ∂τ̄ − τ∂τ + σ∂σ − σ̄∂σ̄, X5 = τ∂σ̄ − σ∂τ̄ , X̄5 = τ̄ ∂σ − σ̄∂τ X6 = z2∂2 + z̄2∂2̄ + u∂u, Xa = a ( z1, z2, τ̄ , σ ) ∂u, Xā = ā ( z̄1, z̄2, τ, σ̄ ) ∂u, (3.1) Xc = cz1∂2 − cz2∂1 + (τcσ − σ̄cτ̄ )∂u, Xc̄ = c̄z̄1∂2̄ − c̄z̄2∂1̄ + (τ̄ c̄σ̄ − σc̄τ )∂u, Xf = f(τ, σ, τ̄ , σ̄)∂u, where a, ā, c = c ( z1, z2, τ̄ , σ ) , c̄ = c̄ ( z̄1, z̄2, τ, σ̄ ) are arbitrary functions and f(τ, σ, τ̄ , σ̄) sa- tisfies the equation fτ τ̄ + fσσ̄ = 0. We note that obvious translational symmetry generators ∂1, ∂1̄, ∂2 and ∂2̄ are particular cases of the generators Xc and Xc̄. We have to emphasize that the subscripts of X designate different vector fields, contrary to our previous convention that subscripts denote partial derivatives. We specify two symmetries from (3.1) for a symmetry reduction of the extended system XI = ∂τ − ∂1, X̄I = ∂τ̄ − ∂1̄. (3.2) Partner Symmetries, Group Foliation and ASD Ricci-Flat Metrics 7 Solutions of CMA invariant with respect to symmetries (3.2) are determined by the conditions uτ = u1, uτ̄ = u1̄. (3.3) Using (3.3), we eliminate uτ and uτ̄ in all the equations (2.8), (2.10) and (2.13) to obtain uσ1̄ = u11̄u12 − u21̄u11, uσ2̄ = u12̄u12 − u22̄u11, (3.4) uσ̄1 = u11̄u1̄2̄ − u2̄1u1̄1̄, uσ̄2 = u1̄2u1̄2̄ − u22̄u1̄1̄, (3.5) u11̄uσσ̄ − u1σ̄u1̄σ = u11u1̄1̄ − u2 11̄. (3.6) We note that equation (3.6) can be obtained by the Legendre transformation v = u− z1u1 − z̄1u1̄, p = −u1, p̄ = −u1̄ of the CMA in new variables vpp̄vσσ̄ − vpσ̄vσp̄ = 1. (3.7) All point symmetry generators of the system of equations CMA, (3.4), (3.5) and (3.6) are listed below X1 = z1∂1 − z̄1∂1̄ − 2 ( z2∂2 − z̄2∂2̄ ) , X2 = z2∂2 + z̄2∂2̄ + u∂u, X3 = a(σ)∂2 + 1 2λ ( z1 )2 a′(σ)∂u, X4 = b(σ̄)∂2̄ + λ 2 ( z̄1 )2 b′(σ̄)∂u, X5 = c′(σ) ( z1∂1 − z2∂2 ) + c(σ)∂σ − 1 2λ ( z1 )2 z2c′′(σ)∂u, X6 = d′(σ̄) ( z̄1∂1̄ − z̄2∂2̄ ) + d(σ̄)∂σ̄ − λ 2 ( z̄1 )2 z̄2d′′(σ̄)∂u, X7 = −λfz2 ( z2, σ ) ∂1 + z1fσ ( z2, σ ) ∂u, X8 = − 1 λ gz̄2 ( z̄2, σ̄ ) ∂1̄ + z̄1gσ̄ ( z̄2, σ̄ ) ∂u, X9 = h ( z2, σ ) ∂u, X10 = k ( z̄2, σ̄ ) ∂u. (3.8) To arrive at the Boyer–Finley equation, we need rotationally invariant solutions of CMA [2]. Among the symmetries (3.8) of our extended system we can choose the symmetry of simultaneous rotations in z1 and z2 complex planes, generated by X1, which is convenient to combine with the point transformation z2 = ep, z̄2 = ep̄. The symmetry generator becomes X = −iX1 = 2i(∂p−∂p̄)− i(z1∂1− z̄1∂1̄), so that new invariant variables are ρ = p+ p̄, q = z1ep/2, q̄ = z̄1ep̄/2 and u = u(ρ, q, q̄, σ, σ̄). After this symmetry reduction the equations CMA, (3.4), (3.5) and (3.6) become respectively uqq̄uρρ − uρquρq̄ = eρ/2, (3.9) uσq̄ = uqq̄(uρq + uq/2)− uqquρq̄, (3.10) uσρ = uρq(uρq + uq/2)− uqquρρ (3.11) together with complex conjugates of (3.10) and (3.11) and, finally, uqq̄uσσ̄ − uσq̄uσ̄q = eρ/2 ( uqquq̄q̄ − u2 qq̄ ) . (3.12) We note that equation (3.12) has the form of the Legendre transform (3.6) of the Monge–Ampère equation in variables q, q̄, σ, σ̄ with ρ playing the role of a parameter, similarly to our remark 8 M.B. Sheftel and A.A. Malykh after equation (3.6). Parameter ρ can be scaled away by changing σ, σ̄ to the new variables s, s̄ defined by s = σeρ/4, s̄ = σ̄eρ/4 when the equation (3.12) becomes uqq̄uss̄ − usq̄us̄q = uqquq̄q̄ − u2 qq̄, which is exactly the Monge–Ampère equation vpp̄vss̄ − vps̄vsp̄ = 1 (3.13) after the Legendre transformation v = u− quq − q̄uq̄, p = −uq, p̄ = −uq̄. (3.14) Therefore, our system contains also transformed Monge–Ampère equation (3.13), though in different variables, in the form (3.12). Our aim is to transform the reduced CMA to the Boyer–Finley equation. On the way to it, we apply one-dimensional Legendre transformation t = uρ, w = u− ρuρ, ρ = −wt, u = w − twt. (3.15) CMA equation (3.9) becomes wqq̄ = −wtte−wt/2. Equations (3.10) and (3.11) take the form wσq̄ = 1 2 wqwqq̄ + wtqe −wt/2, wtσ = 1 2 wqwtq − wqq (3.16) and complex conjugate equations. The image of equation (3.12) with the use of other equations becomes wσσ̄ = wtte −wt + 1 2 ( wq̄wtq + wqwtq̄ − 1 2 wqwq̄wtt ) e−wt/2. (3.17) The final step is to set w = vt, which makes all the equations to be total derivatives with respect to t, and then integrate the equations with respect to t. To simplify the notation, we also change σ to z and σ̄ to z̄. The reduced CMA takes the form of the Boyer–Finley equation vqq̄ = 2e−vtt/2. (3.18) Equations (3.16) together with their complex conjugates read vqq = −vtz + 1 4 v2 tq, (3.19) vq̄q̄ = −vtz̄ + 1 4 v2 tq̄, (3.20) vq̄z = vtqe −vtt/2, (3.21) vqz̄ = vtq̄e −vtt/2, (3.22) and the equation (3.17) becomes vzz̄ = −e−vtt + 1 2 vtqvtq̄e −vtt/2. (3.23) Partner Symmetries, Group Foliation and ASD Ricci-Flat Metrics 9 According to our remark after equation (3.14), our final system contains besides Boyer–Finley equation (3.18) also transformed Monge–Ampère equation, which is a consequence of this system, though its explicit form, being a bit lengthy, is not presented here. Therefore, we have hopes that finding some noninvariant solutions of the Boyer–Finley equation, we will be able to lift them to noninvariant solutions of the complex Monge–Ampère equation, this being our main goal. Point symmetries generators of the above system are X1 = ∂t, X2 = q∂q + q̄∂q̄ + 2t∂t + ( 4v − 2t2 ) ∂v, X3 = qa(z)∂v, X4 = X̄3, X5 = b(z)∂v, X6 = X̄5, X7 = ( tc(z)− q2c′(z)/2 ) ∂v, X8 = X̄7, X9 = d(z)∂q + { q3d′′(z)/3− 2qtd′(z) } ∂v, X10 = X̄9, (3.24) X11 = 1 2 f ′(z)q∂q + f(z)∂z + { q4f ′′′(z)/24− tq2f ′′(z)/2 + t2f ′(z)/2 } ∂v, X12 = X̄11, where the bars mean complex conjugation. All functions are arbitrary and smooth. No nontrivial contact symmetries exist. 4 Operators of invariant differentiation and second-order differential invariants For the group foliation [14, 18, 23] of the system (3.18)–(3.23) we choose the infinite dimen- sional Lie subgroup generated by X11 and X̄11 since it contains maximum number of arbitrary functions and hence maximum number of constraints for differential invariants with respect to this subgroup. In group foliation, an important role is played by operators of invariant differentiation which, by definition, commute with any prolongation of symmetry generatorsX11 and X̄11. The number of independent operators of invariant differentiation is the same as the number of independent variables, that is, five in our case. The equations determining such operators one can find in Ovsiannikov’s book [20] or in our paper [14]. Here we present only the result for solving these equations which fixes the following form of operators of invariant differentiation: δ = Dt, ∆q = qDq, ∆̄q = q̄Dq̄, ∆z = q2(2Dz − vtqDq), ∆̄z = q̄2(2Dz̄ − vtq̄Dq̄), (4.1) where D denotes total derivative with respect to its letter subscript. Operators (4.1), when acting on an invariant, generate again a (differential) invariant, increasing its order by one unit. Invariant differentiations may also generate differential invariants even when acting on a non- invariant quantities (‘pre-invariants’). A basis of differential invariants is formed by invariants such that invariant differentiations can generate any invariant of an arbitrary order by repeated applications to basis invariants. A zeroth-order invariant is t. A single first-order invariant is ω1 = qvq + q̄vq̄ + 2tvt − 4v. The complete set of second-order independent differential invariants consists of 12 invariants. Indeed, the dimension of the Nth prolongation space, where N is the order of the prolongation is νN = n+m (N+n)! N !n! , where n and m are the numbers of independent and dependent variables, respectively, and N = 1, 2, 3, . . . . In our case n = 5, m = 1, so νN = 5+ (N+5)! N !5! , which for N = 2 yields ν2 = 26. The dimension of orbits rN is the rank of the system of Nth prolongations of generators X11 and X̄11, which is equal to the number of arbitrary functions of z, z̄ that they contain. For N = 2, we have arbitrary functions in the second prolongation of our two 10 M.B. Sheftel and A.A. Malykh generators f ′′′′(z), f ′′′′′(z), f̄ ′′′′(z̄), f̄ ′′′′′(z̄) in addition to those which appear in the last line of (3.24), that is, r2 = 12. The dimension of the space of invariants is dimZN = νN − rN and for N = 2 this becomes dimZ2 = ν2 − r2 = 14. Therefore, in addition to the two invariants of zeroth and first-order we must obtain 12 independent second-order invariants. All of them are listed below: ω2 = qq̄e−vtt/2, (4.2) ω3 = ∆q(ω1) = q(qvqq + q̄vqq̄ + 2tvtq − 3vq), (4.3) ω̄3 = ∆̄q(ω1) = q̄(q̄vq̄q̄ + qvqq̄ + 2tvtq̄ − 3vq̄), (4.4) ω4 = δ(ω1) = qvtq + q̄vtq̄ + 2tvtt − 2vt, (4.5) ω5 = qq̄vqq̄ = ∆̄q(qvq) = ∆q(q̄vq̄), (4.6) ω6 = q2q̄(2vq̄z − vqq̄vtq), ω̄6 = qq̄2(2vqz̄ − vqq̄vtq̄), (4.7) ω7 = q2 ( vtz + vqq − v2 tq/4 ) , ω̄7 = q̄2 ( vtz̄ + vq̄q̄ − v2 tq̄/4 ) , (4.8) ω8 = q3q̄3(vqq̄vzz̄ − vqz̄vq̄z), (4.9) ω9 = ∆z(ω1) = q2 { 4tvtz + 2qvqz + 2q̄vq̄z − 8vz − vtq(qvqq + q̄vqq̄ + 2tvtq − 3vq) } , (4.10) ω̄9 = ∆̄z(ω1) = q̄2 { 4tvtz̄ + 2q̄vq̄z̄ + 2qvqz̄ − 8vz̄ − vtq̄(q̄vq̄q̄ + qvqq̄ + 2tvtq̄ − 3vq̄) } . (4.11) 5 Automorphic and resolving equations Here we fix the form of automorphic and resolving equations using only invariant variables. We choose a set of 5 independent invariant variables (same number as in the original system (3.18)– (3.23)) and three remaining differential invariants are considered as new invariant unknowns in the automorphic system (5.2) (see explanation below (5.1)). Integrability conditions of these equations together with the original system yield resolving equations for the unknown functions F , G and Ḡ in (5.2). This task is much simplified by applying our modification of the method which uses commutator algebra of the operators of invariant differentiation together with its Jacobi identities [14, 18, 23]. For any solution of resolving equations for F , G and Ḡ the sys- tem (5.2) possesses the automorphic property: any solution can be obtained from any other solution by a symmetry group transformation generated by X11 and X̄11. All our equations (3.18)–(3.23) can be expressed solely in terms of differential invariants as follows: ω5 = 2ω2 (3.18), ω7 = 0 (3.19), ω̄7 = 0 (3.20), ω6 = 0 (3.21), ω̄6 = 0 (3.22), ω8 = −2ω3 2 (3.23). (5.1) Hence out of the twelve second-order invariants (4.2)–(4.11) there are only six independent ones. Therefore, for the second–order of prolongation we have only eight independent invariants together with t and ω1 on the solution manifold of our system of six equations. For the group foliation, we should separate them into two groups: independent and dependent invariant va- riables. In order not to loose any solutions of our original equations, the number of independent invariant variables should be five, the same as in the original equations. We choose them to be (t, ω1, ω2, ω3, ω̄3). Thus, only three remaining invariants should be chosen as new invariant unknown functions of the five independent invariant variables ω4 = F (t, ω1, ω2, ω3, ω̄3) = δ(ω1), ω9 = G(t, ω1, ω2, ω3, ω̄3) = ∆z(ω1), ω̄9 = Ḡ(t, ω1, ω2, ω3, ω̄3) = ∆̄z(ω1), (5.2) Partner Symmetries, Group Foliation and ASD Ricci-Flat Metrics 11 which is the general form of the automorphic system. Some part of resolving equations we obtain from inner integrability conditions for these three equations together with the equations that follow from the definitions of ω3, ω̄3 ω3 = ∆q(ω1), ω̄3 = ∆̄q(ω1). (5.3) These conditions are obtained by invariant cross differentiation of equations (5.2) and (5.3) where we will use the commutator algebra of operators of invariant differentiation [δ,∆q] = [δ, ∆̄q] = 0, [δ,∆z] = −ωtz∆q, [δ, ∆̄z] = −ω̄tz∆̄q, [∆q,∆z] = 2∆z − ωqz∆q, [∆̄q, ∆̄z] = 2∆̄z − ω̄qz∆̄q, [∆q, ∆̄q] = 0, [∆q, ∆̄z] = ω2ωtt∆̄ q, [∆̄q,∆z] = ω2ωtt∆ q, [∆z, ∆̄z] = 2ω2 ( ω̄tz∆ q − ωtz∆̄q ) . (5.4) Here coefficients are the following third-order invariants ωtt = vttt = δ(vtt), ωtz = qvttq = ∆q(vtt), ω̄tz = q̄vttq̄ = ∆̄q(vtt), (5.5) ωqz = q2vtqq − qvtq = ∆q(qvtq)− 2qvtq = − { q2(vttz − vtqvttq/2) + qvtq } = −1 2 ∆z(vtt)− qvtq, (5.6) ω̄qz = q̄2vtq̄q̄ − q̄vtq̄ = ∆̄q(q̄vtq̄)− 2q̄vtq̄ = − { q̄2(vttz̄ − vtq̄vttq̄/2) + q̄vtq̄ } = −1 2 ∆̄z(vtt)− q̄vtq̄, (5.7) where the alternative expressions for ωqz and ω̄qz in the second lines of (5.6) and (5.7) are obtained by using equations (3.19) and (3.20), respectively. In the integrability conditions of equations (5.2) and (5.3) new third-order invariants ap- pear which are obtained by the action of operators of invariant differentiation on second-order invariants ω2, ω3, ω̄3. We introduce for them the following notation ωit = δ(ωi), ωiq = ∆q(ωi), ωiq̄ = ∆̄q(ωi), ωiz = ∆z(ωi), ωiz̄ = ∆̄z(ωi), (5.8) where i = 2, 3, together with complex conjugates to the equations at i = 3. Equations (5.8) determine projections of operators of invariant differentiation on the space of invariants δ = ∂t + F∂ω1 + ω2t∂ω2 + ω3t∂ω3 + ω̄3t∂ω̄3 , ∆q = ω3∂ω1 + ω2q∂ω2 + ω3q∂ω3 + ω̄3q̄∂ω̄3 , ∆̄q = ω̄3∂ω1 + ω̄2q∂ω2 + ω3q̄∂ω3 + ω̄3q∂ω̄3 , ∆z = G∂ω1 + ω2z∂ω2 + ω3z∂ω3 + ω̄3z̄∂ω̄3 , ∆̄z = Ḡ∂ω1 + ω̄2z∂ω2 + ω3z̄∂ω3 + ω̄3z∂ω̄3 . Then invariant integrability conditions for equations (5.2) and (5.3) can be obtained by applying commutator relations between operators of invariant differentiation (5.4) to the first- order invariant ω1 ∆q(F ) = ω3t, ∆̄q(F ) = ω̄3t, ∆z(F ) = δ(G) + ω3ωtz, ∆̄z(F ) = δ(Ḡ) + ω̄3ω̄tz, ∆q(G) = 2G+ ω3z − ω3ωqz, ∆̄q(Ḡ) = 2Ḡ+ ω̄3z − ω̄3ω̄qz, ∆̄q(G) = ω̄3z̄ − ω3ωqz̄, ∆q(Ḡ) = ω3z̄ − ω̄3ωqz̄, ∆̄z(G) = ∆z(Ḡ) + 2ω2(ω̄3ωtz − ω3ω̄tz) (5.9) with the reality conditions ω̄3q̄ = ω3q̄ and ω̄qz̄ = ωqz̄. 12 M.B. Sheftel and A.A. Malykh These equations have already the form of resolving equations if we consider here third-order invariants defined in (5.8) as auxiliary unknowns which are functions of t, ω1, ω2, ω3, ω̄3. Then we must add new resolving equations following from their definitions (5.8). Some of them we obtain by applying commutator relations (5.4) between invariant differentiations to independent second-order invariant variables ω2, ω3, ω̄3 as follows ∆q(ωit) = δ(ωiq), ∆̄q(ωit) = δ(ω̄iq), ∆̄q(ωiq) = ∆q(ω̄iq), ∆z(ωit) = δ(ωiz)− ωtzωiq, ∆̄z(ω̄it) = δ(ω̄iz)− ω̄tzω̄iq, ∆q(ωiz) = ∆z(ωiq) + 2ωiz − ωqzωiq, ∆̄q(ω̄iz) = ∆̄z(ω̄iq) + 2ω̄iz − ω̄qzω̄iq, ∆q(ω̄iz) = ∆̄z(ωiq)− ωqz̄ω̄iq, ∆̄q(ωiz) = ∆z(ω̄iq)− ωqz̄ωiq, ∆z(ω̄iz) = ∆̄z(ωiz) + 2ω2(ω̄tzω2q − ωtzω̄2q), (5.10) where i = 2, 3, plus complex conjugate equations at i = 3. Here ω̄2t = ω2t. To obtain further resolving equations, we consider Jacobi identities between triples of ope- rators of invariant differentiation using commutator relations (5.4) ∆q(ωtz) = δ(ωqz) + 2ωtz, ∆̄q(ω̄tz) = δ(ω̄qz) + 2ω̄tz, ∆̄q(ωtz) = ∆q(ω̄tz) = −ω2δ(ωtt)− ω2tωtt, ∆z(ω̄tz) = 2ω2δ(ωtz) + ωtz(2ω2t − ω2ωtt), ∆̄z(ωtz) = 2ω2δ(ω̄tz) + ω̄tz(2ω2t − ω2ωtt), ∆q(ω̄qz) = −ω2∆̄q(ωtt) + ωtt(2ω2 − ω̄2q), ∆̄q(ωqz) = −ω2∆q(ωtt) + ωtt(2ω2 − ω2q), ∆z(ω̄qz) = 2ω2∆̄q(ωtz)− 2ωtz(2ω2 − ω̄2q) + ω2 2ω 2 tt, ∆̄z(ωqz) = 2ω2∆q(ω̄tz)− 2ω̄tz(2ω2 − ω2q) + ω2 2ω 2 tt, ω2 { ∆z(ωtt) + 2∆q(ωtz) } = −ωtt(ω2z + ω2ωqz)− 2ωtz(2ω2 + ω2q), ω2 { ∆̄z(ωtt) + 2∆̄q(ω̄tz) } = −ωtt(ω̄2z + ω2ω̄qz)− 2ω̄tz(2ω2 + ω̄2q). (5.11) For example, the first equation in (5.11) is obtained from the Jacobi identity [[δ,∆q],∆z] + [[∆q,∆z], δ]+ [[∆z, δ],∆q] = 0, while the third and fourth equations are obtained from the single Jacobi identity [[δ,∆z], ∆̄z] + [[∆z, ∆̄z], δ] + [[∆̄z, δ],∆z] = 0. Still more resolving equations follow from the definitions (5.5)–(5.7) of the third-order invari- ants that appear as coefficients of commutator algebra of invariant differentiations. Obvious consequences are obtained by invariant cross differentiations of each pair of the three equations in (5.5) ∆q(ωtt) = δ(ωtz), ∆̄q(ωtt) = δ(ω̄tz), ∆̄q(ωtz) = ∆q(ω̄tz). (5.12) Invariant cross differentiation of the first equation in (5.5) and (5.6), (5.7) written in the form ∆z(vtt) = −2ωqz − 2qvtq, ∆̄z(vtt) = −2ω̄qz − 2q̄vtq̄ (5.13) yields ∆z(ωtt) = −2δ(ωqz) + ω2 tz − 2ωtz, ∆̄z(ωtt) = −2δ(ω̄qz) + ω̄2 tz − 2ω̄tz. (5.14) Invariant cross differentiations of the second and third equations in (5.5) together with (5.6) and (5.7), respectively, set in the form δ(qvtq) = ωtz and ∆q(qvtq) = ωqz + 2qvtq, together with complex conjugate equations reproduce first two equations in (5.11). Invariant cross differentiations of the second and third equations in (5.5) together with equa- tions (5.6) and (5.7), respectively, taken in the form (5.13), yield ∆z(ωtz) = −2∆q(ωqz) + ωqz(ωtz + 2), ∆̄z(ω̄tz) = −2∆̄q(ω̄qz) + ω̄qz(ω̄tz + 2). (5.15) Partner Symmetries, Group Foliation and ASD Ricci-Flat Metrics 13 Invariant cross differentiations of the second equation in (5.5) and equation (5.7), taken in the form (5.13), and also of the third equation in (5.5) together with (5.6) in the form (5.13) yield ∆̄z(ωtz) = −2∆q(ω̄qz)− 4ω2t − ω2ωttωtz, ∆z(ω̄tz) = −2∆̄q(ωqz)− 4ω2t − ω2ωttω̄tz. (5.16) Finally, the invariant cross differentiation of equations (5.6) and (5.7) taken in the form (5.13) yields the last resolving equation ∆̄z(ωqz) = ∆z(ω̄qz) + 2ω2(ωtz − ω̄tz). (5.17) Thus, the complete set of resolving equations consists of equations (5.9), (5.10), (5.11), (5.12), (5.14), (5.15), (5.16) and (5.17). 6 Some solutions of the extended system In this section we replace the task of solving the set of resolving equations by a simpler problem of finding particular solutions for the commutator algebra of operators of invariant differentiation [14, 18, 23]. As a result, we find some solutions of the Boyer–Finley equation which look very nontrivial and seem to be new. They also satisfy all other equations (3.18)– (3.23) of the extended system and therefore they also solve the Legendre-transformed CMA equation (3.12). In the next section, we obtain a noninvariant solution of CMA by applying an inverse Legendre transformation to one of the obtained solutions. The extended system (3.18)–(3.23) contains, besides Boyer–Finley equation (3.18) (from now on denoted as BF), the transformed Monge–Ampère equation which is a consequence of this system. Therefore, solutions of this system satisfy simultaneously BF and the transformed CMA and hence provide a lift of solutions of (3.18) to those of CMA. With our choice of the symmetry for the group foliation, non-invariant solutions of the Boyer– Finley equation, obtained in [3, 14] and used in [17] to generate heavenly metrics, cannot be lifted to noninvariant solutions of CMA. Therefore, in order to solve the resolving equations of the group foliation, constructed for the extended system, we have to discover some other solutions of the BF equation (3.18), which by construction will be compatible with all other equations of our system (3.18)–(3.23). However, we have too many resolving equations to solve. Because of that, we use instead the strategy applied in our paper [14] to the single BF equation, namely, to consider the commutator algebra (5.4) of operators of invariant differentiation and make some ansatz simplifying this algebra. The most obvious ansatz is to make as many commutators as possible to vanish ωtt = vttt = 0, ωtz = 0 ⇒ vttq = 0, ω̄tz = 0 ⇒ vttq̄ = 0, (6.1) ωqz = 0 ⇒ qvttz + vtq = 0, ω̄qz = 0 ⇒ q̄vttz̄ + vtq̄ = 0, (6.2) so that the only nonzero commutators are [∆q,∆z] = 2∆z, [∆̄q, ∆̄z] = 2∆̄z. The first equation in (6.1) is very restrictive since it admits only quadratic t-dependence v = α 2 t2 + βt+ γ, (6.3) where α, β, γ are functions of q, q̄, z, z̄. Plugging ansatz (6.3) in other equations of the extended system, we end up with the following solution to all of six equations of this system v = − t 2 2 [ lnκ′(z) + ln κ̄′(z̄) ] + t [ σ(z) + q2 κ ′′(z) 2κ′(z) + σ̄(z̄) + q̄2 κ̄ ′′(z̄) 2κ̄′(z̄) ] 14 M.B. Sheftel and A.A. Malykh + 2qq̄ √ κ′(z)κ̄′(z̄)− κ(z)κ̄(z̄) + q4(3κ′′ 2 − 2κ′κ′′′) 48κ′2 + q̄4(3κ̄′′ 2 − 2κ̄′κ̄′′′) 48κ̄′2 − q2 2 σ′(z)− q̄2 2 σ̄′(z̄) + qν(z) + q̄ν̄(z̄) + ρ(z) + ρ̄(z̄). (6.4) To get more general dependence on t, we skip the first ansatz in (6.1) and also the ansatz (6.2), since the latter does not imply the vanishing of the commutators [∆q,∆z] and its complex conjugate. Thus, we now make the only ansatz vttq = 0 and vttq̄ = 0. Then, without making any more assumptions, we obtain the two following solutions to the whole extended system: v = −(t+ C)2 [ ln (t+ C)− 1 2 ] − t2 [ 1 2 ln (c1 + c̄1)− ln a− 1 ] + t [ 2C ln a− (d− d̄)2 4a + ϕ0 + ϕ̄0 ] + C2 ln a− C(d− d̄)2 4a + ψ0 + ψ̄0 + (t+ C) { 2 √ c1c̄1qq̄ a − q2 c1 a + q [ d′ √ c1 − √ c1(d− d̄) a ] − q̄2 c̄1 a + q̄ [ d̄′√ c̄1 − √ c̄1(d− d̄) a ]} − q3 6 d′′ √ c1 + q2 2 ( d′ 2 4c1 − 2Cc1 a − ϕ′0 ) − q̄3 6 d̄′′√ c̄1 + q̄2 2 ( d̄′ 2 4c̄1 − 2Cc̄1 a − ϕ̄′0 ) + qρ1 + q̄ρ̄1, (6.5) where C 6= 0 is a real constant, a = ā = c̄1z + c̄1z̄ + c0 with constant c1, c̄1, c0, d = d(z), d̄ = d̄(z̄), ρ1 = ρ1(z), ρ̄1 = ρ̄1(z̄), ϕ0 = ϕ0(z), ϕ̄0 = ϕ̄0(z̄), ψ0 = ψ0(z), ψ̄0 = ψ̄0(z̄) are arbitrary functions and the primes denote derivatives of functions of a single variable. The second solution obtained with the same ansatz has the form v = −t2 [ ln t+ 1 2 (ln a′ + ln ā′)− ln (a+ ā)− 3 2 ] + t { 2qq̄ √ a′ā′ a+ ā + q2 ( a′′ 2a′ − a′ a+ ā ) + q̄2 ( ā′′ 2ā′ − ā′ a+ ā ) + q [ d′√ a′ − √ a′(d− d̄) a+ ā ] + q̄ [ d̄′√ ā′ + √ ā′(d− d̄) a+ ā ] − (d− d̄)2 4(a+ ā) + ϕ0 + ϕ̄0 } + q4 ( a′′ 2 16a′ 2 − a′′′ 24a′ ) + q3 6 ( a′′d′ a′ 3/2 − d′′√ a′ ) + q2 ( d′2 8a′ − ϕ′0 2 ) + qρ1 + ψ0 + q̄4 ( ā′′ 2 16ā′ 2 − ā′′′ 24ā′ ) + q̄3 6 ( ā′′d̄′ ā′ 3/2 − d̄′′√ ā′ ) + q̄2 ( d̄′ 2 8ā′ − ϕ̄′0 2 ) + q̄ρ̄1 + ψ̄0, (6.6) where a = a(z), ā = ā(z̄), d = d(z), d̄ = d̄(z̄), ρ1 = ρ1(z), ρ̄1 = ρ̄1(z̄), ϕ0 = ϕ0(z), ϕ̄0 = ϕ̄0(z̄), ψ0 = ψ0(z), ψ̄0 = ψ̄0(z̄) are all arbitrary functions of a single variable. It is interesting to note that the functions a and ā come from the general solution ε(z, z̄) = √ a′(z)ā′(z̄) a+ ā of the Liouville equation (ln ε2)zz̄ = 2ε2. We note that in the process we have obtained some solutions (6.4), (6.5), and (6.6) of the Boyer–Finley equation in the form (3.18), which seem to be new. We also mention separable solutions of the Boyer–Finley equation, which reduce in essence to solutions of the Liouville equation, obtained by Tod [26] and recently by Dunajski et al. [5]. Partner Symmetries, Group Foliation and ASD Ricci-Flat Metrics 15 These solutions turn out to be invariant with respect to one-parameter subgroup of the conformal symmetry group of the Boyer–Finley equation, so it generates a metric with at least two Killing vectors, one of which comes from the BF equation, being itself a rotational symmetry reduction of CMA. We see that the Liouville equation arises also in the present paper but in a different context, not implying separability of our solutions. 7 Simultaneous solutions to Boyer–Finley and complex Monge–Ampère equations We study here the most nontrivial solution (6.6) of the extended system. It should be trans- formed to a simultaneous solution u of the Legendre-transformed CMA (3.12) and Boyer– Finley (3.9) equations together with other equations (3.10) and (3.11) of the extended system. To achieve this, we apply to solution (6.6) one-dimensional Legendre transformation contained in formula (3.15): ρ = −wt, u = w − twt together with w = vt to obtain t = (a+ ā)√ a′ā′ eρ/2 and the solution becomes u = 2(a+ ā)√ a′ā′ eρ/2 + 2qq̄ √ a′ā′ (a+ ā) + q2 ( a′′ 2a′ − a′ a+ ā ) + q̄2 ( ā′′ 2ā′ − ā′ a+ ā ) + q [ d′√ a′ − √ a′(d− d̄) a+ ā ] + q̄ [ d̄′√ ā′ + √ ā′(d− d̄) a+ ā ] − (d− d̄)2 4(a+ ā) + ϕ0 + ϕ̄0, (7.1) where we change the notation of independent variables from z, z̄ to σ, σ̄ for the functions a, ā, d, d̄, ϕ0, ϕ̄0. The expression (7.1) is a solution of the Legendre-transformed CMA (3.12). To transform back (7.1) to a solution of the CMA equation (3.13), which after changing the notation v 7→ Ω and setting s = σeρ/4, s̄ = σ̄eρ/4 becomes Ωpp̄Ωσσ̄ − Ωpσ̄Ωσp̄ = eρ/2 (7.2) we apply to solution (7.1) the inverse Legendre transformation of the form (3.14) Ω = u− quq − q̄uq̄, p = −uq, p̄ = −uq̄. (7.3) To eliminate q, q̄ from the solution, the two latter equations in (7.3) can easily be solved for q and q̄ q = αp+ βp̄+ γ, q̄ = ᾱp̄+ βp+ γ̄, where α = A ∆ , ᾱ = Ā ∆ , β = β̄ = B ∆ , γ = C ∆ , γ̄ = C̄ ∆ , A = ā′[2a′2 − a′′(a+ ā)], Ā = a′[2ā′2 − ā′′(a+ ā)], (7.4) B = B̄ = 2(a′ā′)3/2, C = 1√ a′ (d′Ā+ a′D̄), C̄ = 1√ ā′ (d̄′A+ ā′D), D = ā′[2a′d′ − a′′(d− d̄)], D̄ = a′[2ā′d̄ ′ + ā′′(d− d̄)], ∆ = a′′ā′′(a+ ā)− 2a′′ā′2 − 2ā′′a′2. 16 M.B. Sheftel and A.A. Malykh Then according to the formula (7.3) for Ω, solution (7.1) finally becomes Ω = ᾱ 2 p2 + α 2 p̄2 + βpp̄+ γp+ γ̄p̄+ ∆(αγ2 + ᾱγ̄2 − 2βγγ̄) 2a′ā′(a+ ā) − (d− d̄)2 4(a+ ā) + 2(a+ ā)√ a′ā′ eρ/2 + ϕ0 + ϕ̄0. (7.5) 8 Anti-self-dual Ricci-f lat metric of Euclidean signature It is well known [21] that solutions of the CMA equation (3.13) govern the Kähler metric ds2 = 2(vpp̄dpdp̄+ vps̄dpds̄+ vsp̄dsdp̄+ vss̄dsds̄), (8.1) which is anti-self-dual (ASD) Einstein vacuum (Ricci-flat) metric with Euclidean signature. The transformation s = σeρ/4, s̄ = σ̄eρ/4, which maps equation (3.13) into equation (7.2), does not change the metric (8.1) and hence the solution (7.5) of the equation (7.2) can be used as a potential of the Kähler metric ds2 = 2(Ωpp̄dpdp̄+ Ωpσ̄dpdσ̄ + Ωσp̄dσdp̄+ Ωσσ̄dσdσ̄). (8.2) Plugging our solution (7.5) into the metric (8.2) we obtain an explicit Einstein vacuum metric with Euclidean signature ds2 = 2βdpdp̄+ 2(pᾱσ̄ + p̄βσ̄ + γσ̄)dpdσ̄ + 2(p̄ασ + pβσ + γ̄σ)dp̄dσ + { p2ᾱσσ̄ + p̄2ασσ̄ + 2pp̄βσσ̄ + 2pγσσ̄ + 2p̄γ̄σσ̄ + [ ∆(αγ2 + ᾱγ̄2 − 2βγγ̄) a′ā′(a+ ā) − (d− d̄)2 2(a+ ā) ] σσ̄ − 2 β eρ/2 } dσdσ̄. (8.3) From the definitions (7.4) of the metric coefficients in (8.3), we see that the only singularities of the metric in a bounded domain are zeros of the denominator ∆ = a′′ā′′(a+ ā)− 2a′′ā′2 − 2ā′′a′2 = 0. (8.4) Indeed, the existence condition for our solution is a′(σ) · ā′(σ̄) 6= 0, so that a and ā are not constant and hence a(σ) + ā(σ̄) 6= 0. The general solution for the singularity condition (8.4) in the case of a′′ · ā′′ 6= 0 is a = iλ− 1 a1σ + a0 , ā = −iλ− 1 ā1σ̄ + ā0 (8.5) and if a′′ = ā′′ = 0, we have a = a1σ + a0, ā = ā1σ̄ + ā0. (8.6) Here λ and a0, a1 are real and complex constants, respectively. Thus, avoiding the choices (8.5) and (8.6) for a and ā, we have the metric (8.3) free of singularities in a bounded domain. To compute Riemann curvature two-forms, we choose the tetrad coframe [7] to be e1 = 1 Ωpp̄ (Ωpp̄dp+ Ωzp̄dz), e2 = Ωpp̄dp̄+ Ωpz̄dz̄ e3 = eρ/2 Ωpp̄ dz, e4 = dz̄, so that the metric (8.2) with the aid of equation (7.2) takes the form ds2 = 2 ( e1e2 +e3e4 ) , where we plug in the solution (7.5) for Ω. Partner Symmetries, Group Foliation and ASD Ricci-Flat Metrics 17 Using the computer algebra package EXCALC under REDUCE [22], we calculated Rieman- nian curvature two-forms [7] for the metric (8.3) R1 1 = 2e−ρ/2 \|a′\|5 ∆3 ∣∣∣2a′′′a′ − 3(a′′)2 ∣∣∣2(e1 ∧ e2 − e3 ∧ e4 ) = −R2 2 = −R3 3 = R4 4, R1 2 = R1 4 = R2 1 = R2 3 = R3 2 = R3 4 = R4 1 = R4 3 = 0 (8.7) with the remaining curvature two-forms being too lengthy for presentation here. For example R1 3 = (ā′)3e−ρ√ a′∆3 { a′′(4a′′′a′ − 3a′′2)(5∆ + 18a′2ā′′) − 12a′2a′′′2[(a+ ā)ā′′ − 2ā′2] + 4a′2a′′′′∆ } e2 ∧ e3 − 2e−ρ/2 a′2 √ ā′ ∆3 \|2a′a′′′ − 3a′′2\|2e1 ∧ e4. (8.8) We see that the only singularities of the curvature in a bounded domain are poles at ∆ = 0 together with a′ ·ā′ = 0 which are the same as those of the metric. We note that our solution (7.5) does not exist if the singularities conditions are satisfied. We observe that though the metric (8.3) contains p and p̄, the curvature is independent of these variables. This is due to the quadratic dependence of the metric on p and p̄. Therefore, the curvature components do not vanish outside of a bounded domain, so our metric is not of an instanton type. One may wonder if there are choices of a(σ), ā(σ̄) for which all components of the Riemannian tensor vanish, so that our solution describes a flat space. The inspection of our result for the Riemannian curvature two-forms (8.7), (8.8) gives the conditions for zero curvature to be of the form a′′′ = 3a′′2 2a′ , ā′′′ = 3ā′′2 2ā′ , which yield a = − 4 k(kσ + l) , ā = − 4 k̄(k̄σ̄ + l̄) , where k, l, k̄, l̄ are arbitrary constants. This result coincides with the singularity condition (8.5) at λ = 0 for which our solution (7.5) does not exists and therefore this is not allowed, so that we will never have a flat space for any allowed choice of a, ā. We have succeeded in converting the solution (7.1) into solution (7.5) of CMA equation (7.2) because of the quadratic dependence of solution (7.1) on q, q̄, which allowed us to eliminate these variables in terms of p, p̄ by solving a linear system. For a more general dependence of solutions on q, q̄, this transition could happen to be impossible to be performed explicitly. In this case, we need to transform Kähler metric (8.1) to the one that involves a solution u of the Legendre-transformed CMA (3.12) (solution (7.1) in our example) by applying the inverse Legendre transformation (3.14) with v replaced by Ω to the metric (8.2). The resulting metric has the form ds2 = 2 ∆− { u2 qq̄ ( uqqdq 2 + uq̄q̄dq̄ 2 ) + ∆+uqq̄dqdq̄ + uqqu 2 q̄zdz 2 + uq̄q̄u 2 qz̄dz̄ 2 + (∆−uzz̄ + 2uqq̄uqz̄uq̄z)dzdz̄ + 2uqq̄(uqquq̄zdqdz + uq̄q̄uqz̄dq̄dz̄) + ∆+(uqz̄dqdz̄ + uq̄zdq̄dz) } , (8.9) where ∆− = uqquq̄q̄ − u2 qq̄ and ∆+ = uqquq̄q̄ + u2 qq̄. Metric (8.9) is again anti-self-dual and Ricci-flat with Euclidean signature for any solution u of the transformed CMA equation (3.12). 18 M.B. Sheftel and A.A. Malykh 9 Invariant and noninvariant solutions We are interested here in ASD Ricci-flat metrics of Euclidean signature that do not admit any Killing vectors. This implies solutions of CMA equation for the Kähler potential of the metric to be noninvariant solutions of CMA. This means that we allow only solution manifolds noninvariant under point symmetries of CMA equation. For a set of solutions of the Boyer–Finley equation we presented such an analysis in detail in [14]. In our case, for CMA equation (7.2), dependent of an extra parameter ρ, we have the following generators of one-parameter point symmetries subgroups Xa1 = a1(ρ)(4∂ρ + Ω∂Ω), Yb = b(ρ)(p∂p + p̄∂p̄ + Ω∂Ω), Zc1 = ic1(ρ)(σ∂σ − σ̄∂σ̄), Vg = gp∂σ − gσ∂p, V̄ḡ = ḡp̄∂σ̄ − ḡσ̄∂p̄, Wh = h∂Ω, W̄h̄ = h̄∂Ω, (9.1) where g = g(p, σ, ρ), ḡ = ḡ(p̄, σ̄, ρ) and h = h(p, σ, ρ), h̄ = h̄(p̄, σ̄, ρ) are arbitrary holomorphic and anti-holomorphic functions of two complex and one real variable, a1(ρ), b(ρ), and c1(ρ) are arbitrary real-valued functions of a single variable. We present here the table of commutators of the symmetry generators where the value of the commutator of the generators standing at the ith row and at the jth column is given at the intersection of the ith row and jth column. The primes denote derivatives of functions of a single variable ρ. Table 1. Commutators of point symmetries of parameter-dependent CMA. Xa1 Yb Zc1 Vg V̄ḡ Wh W̄h̄ Xa1 0 4Ya1b′ 4Za1c′1 4Va1gρ 4V̄a1ḡρ 4Wa1hρ 4W̄a1h̄ρ Yb 0 0 Vb(pgp−g) V̄b(p̄ḡp̄−ḡ) Wb(php−h) W̄b(p̄h̄p̄−h̄) Zc1 0 iVc1(σgσ−g) −iV̄c1(σ̄ḡσ̄−ḡ) iWc1σhσ −iW̄c1σ̄h̄σ̄ Vg 0 0 WVg(h) 0 V̄ḡ 0 0 W̄V̄ḡ(h̄) Wh 0 0 W̄h̄ 0 For any solution Ω = f(p, p̄, σ, σ̄, ρ) of equation (7.2), the condition for this solution to be invariant under a generator X of an arbitrary one-parameter symmetry subgroup of (7.2) has the form X(f − Ω)\|Ω=f = 0. (9.2) With X equal to a linear combination of the generators (9.1) with arbitrary constant coefficients (in our case absorbed by arbitrary functions in the generators), the invariance condition (9.2) for the solution Ω = f(p, p̄, σ, σ̄, ρ) becomes gpfσ − gσfp + ḡp̄fσ̄ − ḡσ̄fp̄ + b(ρ)(pfp + p̄fp̄) + ic̃(ρ)(σfσ − σ̄fσ̄) + 4ã(ρ)fρ − (ã(ρ) + b(ρ))f − h− h̄ = 0. (9.3) To check the non-invariance of the solution (7.5), one should plug f equal to the right-hand side of (7.5) in the condition (9.3) and determine either a contradiction or some special forms of the coefficient functions in (7.5) which should be avoided for a noninvariant solution. Partner Symmetries, Group Foliation and ASD Ricci-Flat Metrics 19 To simplify the invariance condition, we have to use optimal Lie subalgebras instead of the general one-dimensional subalgebra used in (9.3) with the generator X = Xã(ρ) + Yb(ρ) + Zc̃(ρ) + Vg + V̄ḡ +Wh + W̄h̄. (9.4) For this purpose, we study the adjoint group actions on one-dimensional Lie subalgebras [19]. Here we must distinguish two cases. Case I: ã(ρ) 6= 0. In this case, we use the adjoint group actions Ad ( exp ( −1 4 Ye ∫ dρ/ã )) (Xã) = Xã − Yb, Ad ( exp ( −1 4 Ze ∫ (c̃/ã)dρ )) (Xã) = Xã − Zc̃ to eliminate Yb and Zc̃ in (9.4), so that the optimal subalgebra becomes X = Xã + Vg + V̄ḡ +Wh + W̄h̄ which results in setting b = c̃ = 0 in the invariance condition (9.3): gpfσ − gσfp + ḡp̄fσ̄ − ḡσ̄fp̄ + ã(ρ)(4fρ − f)− h− h̄ = 0. (9.5) Case II: ã(ρ) = 0. Then using the adjoint group actions Ad(exp(εWh) (Yb) = Yb + εWb(php−h), Ad(exp(εVg) (Yb) = Yb + εVg̃ together with their complex conjugates, we can eliminate Vg, Wh, V̄ḡ, W̄h̄ from (9.4) at ã(ρ) = 0 and the second optimal one-dimensional subalgebra becomes X = Yb + Zc̃. (9.6) The invariance condition (9.3) in the Case II due to the result (9.6) for the optimal subalgebra implies g = ḡ = ã = h = h̄ = 0, so that the invariance condition (9.3) becomes b(ρ)(pfp + p̄fp̄ − f) + ic̃(ρ)(σfσ − σ̄fσ̄) = 0. (9.7) After some routine computations we discover that in both Cases I and II our solution (7.5) generically does not satisfy the invariance conditions (9.5) and (9.7), respectively, and hence it is noninvariant in the generic case, that is, with no restrictions on arbitrary functions of one variable in solution (7.5). A full classification of particular choices of the functional parameters that correspond to invariant solutions presents a difficult problem which is still expecting its solution. 10 Conclusion The problem of obtaining explicitly the metric of K3 gravitational instanton or at least some pieces of it, which will not admit any Killing vectors (no continuous symmetries), has motivated our search for non-invariant solutions to the elliptic complex Monge–Ampère equation. In recent years we have developed three approaches to the latter problem: partner symmetries, that is, invariance with respect to a certain nonlocal symmetry, symmetry reduction with respect to symmetry group parameters introduced explicitly in the theory as new independent variables and a version of the group foliation method, which is based on solving commutator algebra relations for operators of invariant differentiation. In this paper, we have combined all these approaches by introducing explicitly symmetry group parameters into the extended system of six PDEs, which determine partner symmetries of CMA, performing symmetry reductions of these 20 M.B. Sheftel and A.A. Malykh equations with respect to the group parameters and, finally, applying the group foliation to the reduced system. Since the final reduced system contains the Boyer–Finley equation together with CMA, though not in the same variables, a solution to the extended system provides a lift from some solutions of the elliptic BF equation to noninvariant solutions of CMA, that is, from rotationally invariant to noninvariant solutions of CMA. To provide an example of our solution procedure, we have chosen the most obvious ansatzes simplifying the commutator algebra of operators of invariant differentiation and obtained some solutions to our extended system which, after Legendre transformations, became new simul- taneous solutions to a parameter-dependent CMA equation and the BF equation. Using the most general of the obtained solutions, we obtained an anti-self-dual Ricci-flat Einstein–Kähler metric with Euclidean signature and computed Riemannian curvature two-forms. The only sin- gularities of the metric and the curvature, located in a bounded domain, exist only for a very special choice of arbitrary functions of one variable in our solution and therefore they can easily be avoided. Considering in detail the conditions for our solution to be invariant under optimal symmetry subgroups of CMA, we have proved that this is a noninvariant solution in the generic case (that is, with no special restrictions on functional parameters) and hence our metric does not admit any Killing vectors. Our main goal here was to demonstrate how our methods may yield ASD Ricci-flat metrics without Killing vectors, for which purpose we have chosen simplest possible non-invariant solu- tions of CMA. Therefore, it is not surprising that our ansatz for the solution was too restrictive to obtain an instanton metric, so that the curvature is not concentrated in a bounded domain. 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USA 74 (1977), 1798–1799. http://dx.doi.org/10.1088/0305-4470/36/39/304 http://arxiv.org/abs/math-ph/0305037 http://dx.doi.org/10.1088/0305-4470/37/30/010 http://arxiv.org/abs/math-ph/0403020 http://dx.doi.org/10.1088/1751-8113/40/31/014 http://arxiv.org/abs/0704.3335 http://dx.doi.org/10.3842/SIGMA.2011.043 http://arxiv.org/abs/1005.0153 http://dx.doi.org/10.1088/0305-4470/34/43/310 http://arxiv.org/abs/math-ph/0108004 http://dx.doi.org/10.1007/BF01218161 http://dx.doi.org/10.1007/BF01218161 http://arxiv.org/abs/gr-qc/0105088 http://dx.doi.org/10.1088/0305-4470/34/1/311 http://dx.doi.org/10.1007/978-1-4684-0274-2 http://dx.doi.org/10.1063/1.522505 http://dx.doi.org/10.1140/epjb/e2002-00286-6 http://dx.doi.org/10.2991/jnmp.2008.15.s3.37 http://arxiv.org/abs/0802.1463 http://dx.doi.org/10.1088/1751-8113/42/39/395202 http://arxiv.org/abs/0904.2909 http://dx.doi.org/10.1088/0264-9381/12/6/018 http://arxiv.org/abs/gr-qc/0105088 http://dx.doi.org/10.1007/BF02418390 1 Introduction 2 Basic equations 3 Reduction of partner symmetries system for CMA 4 Operators of invariant differentiation and second-order differential invariants 5 Automorphic and resolving equations 6 Some solutions of the extended system 7 Simultaneous solutions to Boyer-Finley and complex Monge-Ampère equations 8 Anti-self-dual Ricci-flat metric of Euclidean signature 9 Invariant and noninvariant solutions 10 Conclusion References

Partner Symmetries, Group Foliation and ASD Ricci-Flat Metrics without Killing Vectors

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