pith. sign in

arxiv: 2603.16847 · v1 · submitted 2026-03-17 · 🧮 math-ph · gr-qc· math.FA· math.MP

Solving gravitational field equations by Wiener-Hopf matrix factorisation, and beyond

M. Cristina C\^amara , Gabriel Lopes Cardoso This is my paper

Pith reviewed 2026-05-15 09:19 UTC · model grok-4.3

classification 🧮 math-ph gr-qcmath.FAmath.MP
keywords Einstein field equationsWiener-Hopf factorizationmonodromy matrixintegrable systemsLax pairtwo-dimensional gravityRiemann-Hilbert problemssolution generation
0
0 comments X

The pith

Einstein's two-dimensional field equations are solved exactly by canonical Wiener-Hopf factorization of a monodromy matrix.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that Einstein's field equations reduced to two dimensions can be treated as an integrable system. This treatment yields exact solutions to the equations and their associated Lax pair at the same time through canonical Wiener-Hopf factorization of the monodromy matrix. Recent progress in factorization methods drawn from singular integral equations and Toeplitz operators makes the process explicit. A tau-invariance property of the factorization supplies a new way to generate additional solutions from known ones. The approach matters because it supplies concrete, exact gravitational solutions that are otherwise difficult to construct directly.

Core claim

Viewing the two-dimensional reductions of Einstein's field equations as an integrable system permits the simultaneous derivation of exact solutions to the field equations and their associated Lax pair by means of a canonical Wiener-Hopf factorisation of the monodromy matrix. The tau-invariance property of this factorisation generates additional solutions without further assumptions. Concrete examples illustrate how this yields explicit solutions in gravitational theories, drawing on developments in Wiener-Hopf techniques from the study of singular integral equations and Toeplitz operators.

What carries the argument

Canonical Wiener-Hopf factorisation of the monodromy matrix, which encodes the integrable structure and produces both the gravitational solutions and the solutions to the Lax pair.

If this is right

  • Exact solutions to the gravitational field equations are obtained explicitly.
  • The associated Lax pair is solved at the same time as the field equations.
  • New solutions are generated from existing ones by the tau-invariance property.
  • Advances in Wiener-Hopf techniques from operator theory apply directly to produce gravitational solutions.
  • Interdisciplinary methods combining general relativity, complex analysis, and operator theory become effective for constructing exact solutions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same factorization method may apply to other nonlinear field theories that reduce to integrable systems in two dimensions.
  • Numerical checks of the generated metrics against known exact solutions such as black-hole spacetimes could test the procedure.
  • The approach may link gravitational solution-generating techniques to other Riemann-Hilbert problems arising in physics.
  • Analogous reductions in higher-dimensional or modified gravity models could be solved by the same monodromy-matrix factorization.

Load-bearing premise

The two-dimensional reductions of Einstein's equations admit a monodromy matrix representation that permits canonical Wiener-Hopf factorization, with the tau-invariance property generating new solutions.

What would settle it

A concrete two-dimensional reduction of Einstein's equations in which the monodromy matrix has no canonical Wiener-Hopf factorization that produces a metric satisfying the original field equations.

Figures

Figures reproduced from arXiv: 2603.16847 by Gabriel Lopes Cardoso, M. Cristina C\^amara.

Figure 1
Figure 1. Figure 1: −m < v < m: four distinct choices of contours. Factorising with respect to Γ we obtain, in each case, a canonical WH factorisation, (τ − τ1) (τ + 1/τ1) (τ − τ2) (τ + 1/τ2) = m−(τ ) m+(τ ) , (30) which yields M(ρ, v) =  m−(∞) 0 0 m−1 − (∞)  =  ∆ 0 0 ∆−1  . (31) (i) Case 1: m+(τ ) = τ1 τ2 τ + 1/τ1 τ + 1/τ2 , m−(τ ) = τ2 τ1 τ − τ1 τ − τ2 , ∆ = τ2/τ1 , (32) yielding a solution which, under the change of co… view at source ↗
Figure 2
Figure 2. Figure 2: Curve C in the Weyl coordinates upper half-plane (ρ > 0, v) for the values m = 2, a = 1. The horizontal axis represents v ∈ R, while the vertical axis represents ρ > 0. 5 Beyond Wiener-Hopf factorisation: τ -invariance and solution generation by multiplication The question of existence of a canonical factorisation was addressed in Section 4. However, even when such a factorisation exists, two further quest… view at source ↗
read the original abstract

By viewing Einstein's field equations -- reduced to two dimensions -- as an integrable system, one can simultaneously obtain exact solutions to both the equations themselves and their associated Lax pair via a canonical Wiener-Hopf factorisation of a so-called monodromy matrix. In this article, we review this remarkable interplay between gravitational field equations, integrable systems, Riemann-Hilbert problems, and Wiener-Hopf factorisation theory, with particular emphasis on developments from the past decade enabled by advances in Wiener-Hopf factorisation techniques arising from the study of singular integral equations and Toeplitz operators. Through a variety of concrete examples, we illustrate how Wiener-Hopf factorisation yields explicit, exact solutions to the field equations of gravitational theories, and how its generalisation through a so-called $\tau$-invariance property provides a new solution-generating method. Along the way, we aim to demonstrate the importance of an interdisciplinary approach -- grounded in General Relativity, Complex Analysis, and Operator Theory -- for the study of gravitational field equations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript claims that two-dimensional reductions of Einstein's field equations can be treated as integrable systems whose exact solutions (together with their Lax pairs) are obtained by canonical Wiener-Hopf factorization of an associated monodromy matrix. It reviews the connections among gravitational field equations, integrable systems, Riemann-Hilbert problems and Wiener-Hopf theory, supplies explicit constructions for several known reductions including the Ernst equation, verifies the resulting solutions by direct substitution, and introduces a tau-invariance property that generates new solutions within the same function class.

Significance. If the derivations hold, the work supplies a systematic, operator-theoretic route to exact solutions of integrable 2D reductions of Einstein's equations and demonstrates the utility of recent advances in Wiener-Hopf factorization for gravitational physics. The explicit constructions, direct verifications by substitution, and parameter-free character of the factorization steps constitute genuine strengths that could be adopted by researchers working on exact solutions in general relativity.

minor comments (3)
  1. §4 (Ernst-equation example): the explicit form of the monodromy matrix and the steps of its factorization are described at a high level; reproducing the full matrix entries and the resulting Wiener-Hopf factors in an appendix would improve reproducibility without lengthening the main text.
  2. The tau-invariance section states that the property maps solutions to solutions 'without additional assumptions,' yet the precise function-class restrictions inherited from the original factorization are not restated; a single clarifying sentence would remove ambiguity.
  3. Notation for the spectral parameter and the contour of the Riemann-Hilbert problem varies slightly between the general setup and the concrete examples; a short table of symbols would aid readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the recognition of its significance, and the recommendation for minor revision. No specific major comments appear in the provided referee report, so we have no individual points to address. We will incorporate any minor editorial or presentational suggestions in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via external operator theory

full rationale

The paper reduces 2D Einstein equations to an integrable system possessing a monodromy matrix, then applies canonical Wiener-Hopf factorization (a standard tool from Toeplitz operators and singular integral equations) to produce explicit solutions to both the field equations and the associated Lax pair. Concrete examples for the Ernst equation and related systems are constructed explicitly and verified by direct substitution; the tau-invariance extension is shown to map solutions to solutions inside the same function class using only the factorization properties already established. No step equates a derived quantity to a fitted input by construction, renames a known result, or relies on a self-citation chain whose validity is presupposed rather than independently verified. The central construction therefore remains independent of the present manuscript's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on the assumption that reduced Einstein equations form an integrable system with a factorizable monodromy matrix; no free parameters or new physical entities are introduced in the abstract.

axioms (2)
  • domain assumption Einstein's field equations reduced to two dimensions can be viewed as an integrable system possessing a monodromy matrix.
    This is the foundational premise stated at the start of the abstract.
  • domain assumption The monodromy matrix admits a canonical Wiener-Hopf factorization that simultaneously solves the field equations and the Lax pair.
    Invoked to obtain the exact solutions.

pith-pipeline@v0.9.0 · 5481 in / 1362 out tokens · 60941 ms · 2026-05-15T09:19:46.611753+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

75 extracted references · 75 canonical work pages

  1. [1]

    1981 Solitons and the inverse scattering transform

    Ablowitz MJ, Segur H. 1981 Solitons and the inverse scattering transform. Philadelphia: SIAM

    work page 1981

  2. [2]

    2022 An explicit Wiener–Hopf factorization algorithm for matrix polynomials and its exact realizations within ExactMPF package

    Adukov VM, Adukova NV, Mishuris G. 2022 An explicit Wiener–Hopf factorization algorithm for matrix polynomials and its exact realizations within ExactMPF package. Proc. A.478, no. 2263, Paper No. 20210941, 22 pp

    work page 2022

  3. [3]

    2004 Inverse scattering transform, KdV, and solitons

    Aktosun T. 2004 Inverse scattering transform, KdV, and solitons. In: Ball JA, Helton JW, Klaus M, Rodman L, editors. Current trends in operator theory and its applications. Basel: Birkh¨ auser. p. 1–22

    work page 2004

  4. [4]

    2010 Thirty years of studies of integrable reductions of Einstein’s field equations

    Alekseev GA. 2010 Thirty years of studies of integrable reductions of Einstein’s field equations. In: 12th Marcel Grossmann Meeting on General Relativity. p. 645–666

    work page 2010

  5. [5]

    2020 Weyl metrics and Wiener–Hopf factorization

    Aniceto P, Cˆ amara MC, Cardoso GL, Rossell´ o M. 2020 Weyl metrics and Wiener–Hopf factorization. J. High Energy Phys.5, 124

    work page 2020

  6. [6]

    2006 Rotating attractors

    Astefanesei D, Goldstein K, Jena RP, Sen A, Trivedi SP. 2006 Rotating attractors. J. High Energy Phys.10, 058

    work page 2006

  7. [7]

    2003 Introduction to classical integrable systems

    Babelon O, Bernard D, Talon M. 2003 Introduction to classical integrable systems. Cambridge: Cambridge University Press

    work page 2003

  8. [8]

    1978 Integration of the Einstein equations by the inverse scattering problem technique and the calculation of the exact soliton solutions

    Belinsky VA, Zakharov VE. 1978 Integration of the Einstein equations by the inverse scattering problem technique and the calculation of the exact soliton solutions. Sov. Phys. JETP48, 985–994

    work page 1978

  9. [9]

    1979 Stationary gravitational solitons with axial symmetry

    Belinsky VA, Sakharov VE. 1979 Stationary gravitational solitons with axial symmetry. Sov. Phys. JETP50, 1–9

    work page 1979

  10. [10]

    2013 The factorization problem: some known results and open questions

    B¨ ottcher A, Spitkovsky IM. 2013 The factorization problem: some known results and open questions. In: Advances in harmonic analysis and operator theory. Vol. 229, Oper Theory Adv Appl. Basel: Birkh¨ auser/Springer. p. 101–122

    work page 2013

  11. [11]

    1987 On the geroch group

    Breitenlohner P, Maison D. 1987 On the geroch group. Ann. Inst. H. Poincare Phys. Theor.46, 215

    work page 1987

  12. [12]

    1964 Electromagnetic fields in a homogeneous, nonisotropic universe

    Brill DR. 1964 Electromagnetic fields in a homogeneous, nonisotropic universe. Phys. Rev.133, B845

    work page 1964

  13. [13]

    2002 Explicit Wiener–Hopf factorization and nonlinear Riemann–Hilbert problems

    Cˆ amara MC, dos Santos AF, Carpentier MP. 2002 Explicit Wiener–Hopf factorization and nonlinear Riemann–Hilbert problems. Proc R Soc Edinburgh Sect A.132, no. 1, 45–74

    work page 2002

  14. [14]

    2007 Lax equations, factorization and Riemann–Hilbert problems

    Cˆ amara MC, dos Santos AF, dos Santos PF. 2007 Lax equations, factorization and Riemann–Hilbert problems. Port Math (N.S.)64, no. 4, 509–533

    work page 2007

  15. [15]

    2008 Matrix Riemann–Hilbert problems and factorization on Riemann surfaces

    Cˆ amara MC, dos Santos AF, dos Santos PF. 2008 Matrix Riemann–Hilbert problems and factorization on Riemann surfaces. J. Funct. Anal.255, no. 1, 228–254

    work page 2008

  16. [16]

    2017 Toeplitz operators and Wiener–Hopf factorisation: an introduction

    Cˆ amara MC. 2017 Toeplitz operators and Wiener–Hopf factorisation: an introduction. Concr Oper.4, no. 1, 130–145. 29

    work page 2017

  17. [17]

    2017 A Riemann-Hilbert approach to rotating attractors

    Cˆ amara MC, Cardoso GL, Mohaupt T, Nampuri S. 2017 A Riemann-Hilbert approach to rotating attractors. J. High Energy Phys.6, 123

    work page 2017

  18. [18]

    2024 Generating new gravitational solutions by matrix multiplication

    Cˆ amara MC, Cardoso GL. 2024 Generating new gravitational solutions by matrix multiplication. Proc. R. Soc. A.480, 20230857

    work page 2024

  19. [19]

    2024 Riemann–Hilbert problems, Toeplitz operators and ergosurfaces

    Cˆ amara MC, Cardoso GL. 2024 Riemann–Hilbert problems, Toeplitz operators and ergosurfaces. J. High Energy Phys.6, 027

    work page 2024

  20. [20]

    2025 Factorisation of symmetric matrices and applications in gravitational theories

    Cˆ amara MC, Cardoso GL. 2025 Factorisation of symmetric matrices and applications in gravitational theories. In: Operator Theory: Advances and Applications, vol. 306. p. 157–181. Birkh¨ auser/Springer

    work page 2025

  21. [21]

    2010 Fredholmness of Toeplitz operators and corona problems

    Cˆ amara MC, Diogo C, Rodman L. 2010 Fredholmness of Toeplitz operators and corona problems. J Funct Anal.259, 1273–1299

    work page 2010

  22. [22]

    2018 New gravitational solutions via a Riemann-Hilbert approach

    Cardoso GL, Serra JC. 2018 New gravitational solutions via a Riemann-Hilbert approach. J. High Energy Phys.3, 080

    work page 2018

  23. [23]

    2024 Weyl–Lewis–Papapetrou coordinates, self-dual Yang–Mills equations and the single copy

    Cardoso GL, Mahapatra S, Nagy S. 2024 Weyl–Lewis–Papapetrou coordinates, self-dual Yang–Mills equations and the single copy. J. High Energy Phys.10, 030

    work page 2024

  24. [24]

    2014 Geroch group description of black holes

    Chakrabarty B, Virmani A. 2014 Geroch group description of black holes. J. High Energy Phys.11, 068

    work page 2014

  25. [25]

    1981 Factorisation of matrix functions and singular integral operators

    Clancey KF, Gohberg I. 1981 Factorisation of matrix functions and singular integral operators. In: Operator Theory: Advances and Applications, vol. 3. Basel: Birkh¨ auser

    work page 1981

  26. [26]

    2025 Symmetries of vanishing nonlinear Love numbers of Schwarzschild black holes

    Combaluzier-Szteinsznaider O, Hui L, Santoni L, Solomon AR, Wong SSC. 2025 Symmetries of vanishing nonlinear Love numbers of Schwarzschild black holes. J. High Energy Phys.3, 124

    work page 2025

  27. [27]

    Theory of Hp spaces

    Duren P. Theory of Hp spaces. 2000 New York: Dover Publications

    work page 2000

  28. [28]

    2002 Transformation techniques towards the factorization of non-rational 2×2 matrix functions

    Ehrhardt T, Speck FO. 2002 Transformation techniques towards the factorization of non-rational 2×2 matrix functions. Linear Algebra Appl.353, 53–90

    work page 2002

  29. [29]

    2002 Generalized Weyl solutions

    Emparan R, Reall HS. 2002 Generalized Weyl solutions. Phys. Rev. D.65, 084025

    work page 2002

  30. [30]

    2010 Integrability of five dimensional minimal supergravity and charged rotating black holes

    Figueras P, Jamsin E, Rocha JV, Virmani A. 2010 Integrability of five dimensional minimal supergravity and charged rotating black holes. Class. Quantum Grav.27, 135011

    work page 2010

  31. [31]

    1990 Twistor characterization of stationary axisymmetric solutions of Einstein’s equations

    Fletcher J, Woodhouse N. 1990 Twistor characterization of stationary axisymmetric solutions of Einstein’s equations. In: London Mathematical Society Lecture Note Series. p. 260–282. Cambridge: Cambridge University Press

    work page 1990

  32. [32]

    2016 Introduction to Model Spaces and Their Operators

    Garcia SR, Mashreghi J, Ross WT. 2016 Introduction to Model Spaces and Their Operators. Cambridge: Cambridge University Press. (Cambridge Studies in Advanced Mathematics)

    work page 2016

  33. [33]

    1971 A method for generating solutions of Einstein’s equations

    Geroch RP. 1971 A method for generating solutions of Einstein’s equations. J Math Phys. 12, 918–924

    work page 1971

  34. [34]

    1972 A method for generating new solutions of Einstein’s equation

    Geroch RP. 1972 A method for generating new solutions of Einstein’s equation. 2. J Math Phys.13, 394–404. 30

    work page 1972

  35. [35]

    1992 One-dimensional Linear Singular Integral Equations

    Gohberg IC, Krupnik NI. 1992 One-dimensional Linear Singular Integral Equations. In: Operator Theory: Advances and Applications; vol. 53. Basel: Birkh¨ auser Verlag

    work page 1992

  36. [36]

    2009 Exact Space-Times in Einstein’s General Relativity

    Griffiths JB, Podolsk´ y J. 2009 Exact Space-Times in Einstein’s General Relativity. Cambridge: Cambridge University Press (Cambridge Monographs on Mathematical Physics)

    work page 2009

  37. [37]

    2004 Stationary and axisymmetric solutions of higher-dimensional general relativity

    Harmark T. 2004 Stationary and axisymmetric solutions of higher-dimensional general relativity. Phys Rev D.70, 124002

    work page 2004

  38. [38]

    1980 New large family of vacuum solutions of the equations of general relativity

    Harrison BK. 1980 New large family of vacuum solutions of the equations of general relativity. Phys. Rev. D21, 1695–1697

    work page 1980

  39. [39]

    2016 Kernels of Toeplitz operators

    Hartmann A, Mitkovski M. 2016 Kernels of Toeplitz operators. Contemp. Math.679, 147–177

    work page 2016

  40. [40]

    1981 Proof of a Geroch conjecture

    Hauser I, Ernst FJ. 1981 Proof of a Geroch conjecture. J Math Phys.22, 1051–1063

    work page 1981

  41. [41]

    2007 Banach Spaces of Analytic Functions

    Hoffman K. 2007 Banach Spaces of Analytic Functions. Mineola, NY: Dover Publications

    work page 2007

  42. [42]

    2003 The Riemann-Hilbert problem and integrable systems

    Its AR. 2003 The Riemann-Hilbert problem and integrable systems. Notices AMS.50, 1389

    work page 2003

  43. [43]

    2024 Integrability in perturbed black holes: Background hidden structures

    Jaramillo JL, Lenzi M, Sopuerta CF. 2024 Integrability in perturbed black holes: Background hidden structures. Phys. Rev. D.110, 104049

    work page 2024

  44. [44]

    2013 Inverse scattering and the Geroch group

    Katsimpouri D, Kleinschmidt A, Virmani A. 2013 Inverse scattering and the Geroch group. J. High Energy Phys.2, 011

    work page 2013

  45. [45]

    2014 An inverse scattering formalism for STU supergravity

    Katsimpouri D, Kleinschmidt A, Virmani A. 2014 An inverse scattering formalism for STU supergravity. J. High Energy Phys.3, 101

    work page 2014

  46. [46]

    2014 An inverse scattering construction of the JMaRT fuzzball

    Katsimpouri D, Kleinschmidt A, Virmani A. 2014 An inverse scattering construction of the JMaRT fuzzball. J. High Energy Phys.12, 070

    work page 2014

  47. [47]

    1977 Symmetries of the stationary Einstein-Maxwell field equations

    Kinnersley W. 1977 Symmetries of the stationary Einstein-Maxwell field equations. 1. J Math. Phys.18, 1529–1537

    work page 1977

  48. [48]

    1977 Symmetries of the stationary Einstein-Maxwell field equations

    Kinnersley W, Chitre DM. 1977 Symmetries of the stationary Einstein-Maxwell field equations. 2. J Math. Phys.18, 1538–1542

    work page 1977

  49. [49]

    1978 Symmetries of the stationary Einstein-Maxwell field equations

    Kinnersley W, Chitre DM. 1978 Symmetries of the stationary Einstein-Maxwell field equations. 3. J Math. Phys.19, 1926–1931

    work page 1978

  50. [50]

    1978 Symmetries of the stationary Einstein & Maxwell equations

    Kinnersley W, Chitre DM. 1978 Symmetries of the stationary Einstein & Maxwell equations. 4. Transformations which preserve asymptotic flatness. J Math. Phys.19, 2037–2042

    work page 1978

  51. [51]

    2021 The Wiener–Hopf technique, its generalizations and applications: constructive and approximate methods

    Kisil AV, Abrahams ID, Mishuris G, Rogosin SV. 2021 The Wiener–Hopf technique, its generalizations and applications: constructive and approximate methods. Proc. R. Soc. A 477, 20210533

    work page 2021

  52. [52]

    2023 Special issue: advances in Wiener–Hopf type techniques: theory and applications

    Kisil A, Mishuris G. 2023 Special issue: advances in Wiener–Hopf type techniques: theory and applications. Proc. R. Soc. A.479, 20230210. 31

    work page 2023

  53. [53]

    2005 Ernst equation and Riemann surfaces: Analytical and numerical methods

    Klein C, Richter O. 2005 Ernst equation and Riemann surfaces: Analytical and numerical methods. Lecture Notes in Physics. Berlin: Springer

    work page 2005

  54. [54]

    1995 Separation of variables and Hamiltonian formulation for the Ernst equation

    Korotkin D, Nicolai H. 1995 Separation of variables and Hamiltonian formulation for the Ernst equation. Phys. Rev. Lett.74, 1272–1275

    work page 1995

  55. [55]

    Integrability and Einstein’s equations

    Korotkin D, Samtleben H. Integrability and Einstein’s equations. Invited contribution to the 2nd edition of the Encyclopedia of Mathematical Physics. (arXiv:2311.07900 [gr-qc])

    work page arXiv

  56. [56]

    1987 Factorisation of measurable matrix functions

    Litvinchuk G, Spitkovsky I. 1987 Factorisation of measurable matrix functions. In Operator theory: advances and applications, vol. 25 (ed. G Heinig), 372 pp. Basel, Switzerland: Birkh¨ auser. Translated from Russian by B. Luderer, with a foreword by B. Silbermann

    work page 1987

  57. [57]

    2009 Infinite-dimensional symmetries of two-dimensional coset models coupled to gravity

    Lu H, Perry MJ, Pope CN. 2009 Infinite-dimensional symmetries of two-dimensional coset models coupled to gravity. Nucl. Phys. B806, 656–683

    work page 2009

  58. [58]

    1978 Are the stationary, axially symmetric Einstein equations completely integrable? Phys

    Maison D. 1978 Are the stationary, axially symmetric Einstein equations completely integrable? Phys. Rev. Lett.41, 521–522

    work page 1978

  59. [59]

    1986 Singular integral operators

    Mikhlin S, Pr¨ ossdorf S. 1986 Singular integral operators. Berlin, Germany: Springer. Translated from German by Albrecht B¨ ottcher and Reinhard Lehmann

    work page 1986

  60. [60]

    1986 Kernels of Toeplitz operators

    Nakazi T. 1986 Kernels of Toeplitz operators. J. Math. Soc. Japan.38, no. 4, 607–616

    work page 1986

  61. [61]

    1991 Two-dimensional gravities and supergravities as integrable system

    Nicolai H. 1991 Two-dimensional gravities and supergravities as integrable system. Lect. Notes Phys.396, 231–273

    work page 1991

  62. [62]

    2021 Einstein–Rosen waves and the Geroch group

    Penna RF. 2021 Einstein–Rosen waves and the Geroch group. J. Math. Phys. 62, 082503

    work page 2021

  63. [63]

    1976 Nonlinear gravitons and curved twistor theory

    Penrose R. 1976 Nonlinear gravitons and curved twistor theory. Gen. Relat. Gravit.7, 31–52

    work page 1976

  64. [64]

    1994 Kernels of Toeplitz operators

    Sarason D. 1994 Kernels of Toeplitz operators. In: Basor EL, Gohberg I, editors. Toeplitz operators and related topics. p. 153–164. Basel: Birkh¨ auser

    work page 1994

  65. [65]

    1995 Classical symmetries of some two-dimensional models coupled to gravity

    Schwarz JH. 1995 Classical symmetries of some two-dimensional models coupled to gravity. Nucl. Phys. B.454, 427–448

    work page 1995

  66. [66]

    2003 Exact solutions of Einstein’s field equations

    Stephani H, Kramer D, MacCallum MAH, Hoenselaers C, Herlt E. 2003 Exact solutions of Einstein’s field equations. Cambridge: Cambridge University Press (Cambridge Monographs on Mathematical Physics)

    work page 2003

  67. [67]

    2015 Constructive methods for factorization of matrix-functions

    Rogosin S, Mishuris G. 2015 Constructive methods for factorization of matrix-functions. IMA J. Appl. Math.81, 365–391

    work page 2015

  68. [68]

    2018 Geroch group description of bubbling geometries

    Roy P, Virmani A. 2018 Geroch group description of bubbling geometries. J. High Energy Phys.08, 129

    work page 2018

  69. [69]

    1987 Real and complex analysis

    Rudin W. 1987 Real and complex analysis. New York: McGraw-Hill

    work page 1987

  70. [70]

    1983 Stationary axisymmetric space-times: a new approach

    Ward RS. 1983 Stationary axisymmetric space-times: a new approach. Gen. Relat. Gravit.15, 105–109

    work page 1983

  71. [71]

    1988 The Geroch group and non-Hausdorff twistor spaces

    Woodhouse NMJ, Mason LJ. 1988 The Geroch group and non-Hausdorff twistor spaces. Nonlinearity1, 73–91. 32

    work page 1988

  72. [72]

    1989 Cylindrical gravitational waves

    Woodhouse NMJ. 1989 Cylindrical gravitational waves. Class. Quantum Grav.6, 933–941

    work page 1989

  73. [73]

    1997 Integrability and Einstein’s equations

    Woodhouse N. 1997 Integrability and Einstein’s equations. Banach Cent. Publ.41, no. 1, 221-232

    work page 1997

  74. [74]

    1970 Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media

    Zakharov VE, Shabat AB. 1970 Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. J. Exp. Theor. Phys.34, 62–69

    work page 1970

  75. [75]

    1980 On the integrability of classical spinor models in two-dimensional space-time

    Zakharov V, Mikhailov A. 1980 On the integrability of classical spinor models in two-dimensional space-time. Commun. Math. Phys.74, no. 1, 21-40. 33

    work page 1980