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arxiv: 2605.17843 · v1 · pith:FIQT3JGMnew · submitted 2026-05-18 · 🌀 gr-qc

Potential Space Symmetries in Ernst-like Formulations of Einstein-Maxwell/ModMax-Scalar field Theories

Leonel Bixano , Tonatiuh Matos This is my paper

Pith reviewed 2026-05-20 10:13 UTC · model grok-4.3

classification 🌀 gr-qc
keywords Ernst potentialsEinstein-Maxwell-Scalar fieldsModMax theoryhidden symmetriesEhlers transformationHarrison transformationsstationary axisymmetric solutionsNoether charges
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The pith

The symmetries of the five-dimensional Ernst-like potential space are fully determined for Einstein-Maxwell-scalar and ModMax-scalar theories.

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

The paper establishes the complete set of visible, hidden, sectorial, and discrete symmetries acting on the real potential space with coordinates (f, ε, ψ, χ, κ) in stationary, axisymmetric EMSF and EMMSF theories. A reader would care because these symmetries provide a systematic way to generate new exact solutions, including charged and rotating ones, from simpler seed solutions like harmonic scalar-vacuum Weyl metrics. The work identifies the solvable Lie algebra for visible symmetries and shows how Ehlers transformations handle gravito-rotational parts while Harrison transformations act on electromagnetic sectors. In the frozen ModMax regime, it demonstrates deformations of these transformations and conditions for their coexistence, along with derivations of Noether charges and metric quadratures.

Core claim

In the real potential space (f,ε,ψ,χ,κ), the exact visible symmetries and their solvable Lie algebra are determined. Hidden symmetries are characterized on invariant subspaces with Ehlers acting in the gravito-rotational sector and electric and magnetic Harrison transformations in static electromagnetic sectors. In the frozen EMMSF regime with v=v0, w=w0, EMSF sectorial transformations are deformed, and coexistence of electric and magnetic Harrison transformations requires dw=0 and d[(v²+w²)/w]=0, selecting the frozen ModMax sector.

What carries the argument

The five real coordinates (f, ε, ψ, χ, κ) of the Ernst-like potential formulation, on which the symmetries act as Lie group transformations.

If this is right

  • The affine geodesic energy remains constant in harmonic branches of the A, B, C one-forms, controlling the quadrature for the metric function k.
  • Metric functions ω and A_φ are determined from Noether charges along Killing directions using dual harmonic functions.
  • Frozen-ModMax Harrison transformations generate charged solution branches from harmonic scalar-vacuum Weyl seeds.
  • Ehlers transformations generate the gravito-rotational branches from the same seeds.
  • Discrete maps exchange electric and magnetic Lewis-Weyl-Papapetrou frames via κ ↦ κ^{-1}, ψ ↦ χ, χ ↦ -ψ, ε ↦ ε - ψχ.

Where Pith is reading between the lines

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

  • These symmetries might extend the solution-generating techniques to include scalar field effects in a controlled way for astrophysical modeling.
  • Similar potential space approaches could be applied to other nonlinear electrodynamics theories beyond ModMax.
  • Testing the generated solutions against numerical relativity simulations could validate the symmetry-based constructions for black hole spacetimes.

Load-bearing premise

The field equations of EMSF and EMMSF theories admit a global Ernst-like potential formulation in real coordinates that allows the symmetries to be realized as Lie-group actions.

What would settle it

A explicit counterexample showing that one of the proposed transformations, such as a modified Harrison map, fails to preserve the field equations when applied to a known solution in the potential space would disprove the completeness of the symmetry group.

read the original abstract

We complete the visible, hidden, sectorial, and discrete symmetries of Ernst-like potential spaces in stationary, axisymmetric Einstein-Maxwell-Scalar Field (EMSF) and Einstein-ModMax-Scalar Field (EMMSF) theories. In the real potential space \((f,\epsilon,\psi,\chi,\kappa)\), we determine the exact visible symmetries and their solvable Lie algebra. We characterize the hidden symmetries on invariant subspaces: Ehlers acts in the gravito-rotational sector, while electric and magnetic Harrison transformations act in static electromagnetic sectors. In the frozen EMMSF regime, \(v=v_0,\ w=w_0\), we show how EMSF sectorial transformations are deformed in ModMax theory. We also show that coexistence of electric and magnetic sectorial Harrison transformations imposes \(d w=0\) and \(d[(v^2+w^2)/w]=0\), selecting precisely the frozen ModMax sector. We study the Hamiltonian formulation, Noether charges, and Casimir invariants of the sectorial algebras. In harmonic branches of the \(A,B,C\) one-forms, the affine geodesic energy is constant, so the quadrature for \(k\) is controlled by the affine-geodesic Hamiltonian. The functions \(\omega\) and \(A_\varphi\) follow from Noether charges along the Killing directions of \(\epsilon\) and \(\chi\), and are written using dual harmonic functions. We examine the electric and magnetic Lewis-Weyl-Papapetrou frames and their discrete map, which sends \(\kappa\mapsto\kappa^{-1}\), \(\psi\mapsto\chi\), \(\chi\mapsto-\psi\), and \(\epsilon\mapsto\epsilon-\psi\chi\). Finally, we apply the sectorial transformations to harmonic scalar--acuum Weyl seeds with independent gravitational and scalar harmonics. Frozen-ModMax Harrison maps generate charged branches, while Ehlers generates the gravito-rotational branch. For these solutions we give the final quadratures for \(k\), \(\omega\), and \(A_\varphi\).

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

2 major / 2 minor

Summary. The manuscript claims to complete the visible, hidden, sectorial, and discrete symmetries of Ernst-like potential spaces in stationary, axisymmetric Einstein-Maxwell-Scalar Field (EMSF) and Einstein-ModMax-Scalar Field (EMMSF) theories. In the real potential space (f,ε,ψ,χ,κ), the exact visible symmetries and their solvable Lie algebra are determined. Hidden symmetries are characterized on invariant subspaces, with Ehlers acting in the gravito-rotational sector and electric/magnetic Harrison transformations in static electromagnetic sectors. In the frozen EMMSF regime (v=v0, w=w0), EMSF sectorial transformations are deformed in ModMax theory, and coexistence of electric and magnetic Harrison transformations is shown to impose dw=0 and d[(v²+w²)/w]=0. The work examines the Hamiltonian formulation, Noether charges, and Casimir invariants, derives quadratures using affine geodesic energy and dual harmonic functions, studies the electric/magnetic Lewis-Weyl-Papapetrou frames and their discrete map, and applies the transformations to harmonic scalar-vacuum Weyl seeds to generate charged and gravito-rotational branches with explicit final quadratures for k, ω, and A_φ.

Significance. If the derivations hold, the work provides a systematic extension of the Ernst-potential formalism to EMSF and EMMSF theories, furnishing explicit Lie-algebraic tools and solution-generating transformations. The explicit treatment of Noether charges, Casimir invariants, affine-geodesic control of quadratures, and the discrete frame map supplies concrete, usable machinery for constructing exact solutions from harmonic seeds. These elements strengthen the practical utility of symmetry methods in stationary axisymmetric spacetimes with scalar and nonlinear electromagnetic matter.

major comments (2)
  1. [Abstract and frozen-regime discussion] Abstract and the paragraph on the frozen EMMSF regime: the central claim states that the symmetries are completed for both EMSF and EMMSF, yet the ModMax sectorial deformations and the coexistence of electric and magnetic Harrison transformations are explicitly restricted to the frozen regime (v=v0, w=w0) with the conditions dw=0 and d[(v²+w²)/w]=0 imposed. This limitation is load-bearing for the EMMSF portion of the title and abstract; the manuscript should state upfront whether the potential formulation and Lie-algebra actions extend to generic (non-frozen) ModMax constitutive relations or whether additional obstructions arise.
  2. [Hamiltonian formulation and Noether charges] Section on Hamiltonian formulation and Noether charges: the statement that the affine geodesic energy is constant in harmonic branches of the A,B,C one-forms is used to control the quadrature for k, with ω and A_φ obtained from Noether charges along the Killing directions of ε and χ. It is not shown how these charges relate to the Casimir invariants of the sectorial algebras; an explicit relation or example would confirm that the Hamiltonian approach is consistent with the Lie-algebra structure derived earlier.
minor comments (2)
  1. [Discrete map between frames] The discrete map is stated to send κ↦κ^{-1}, ψ↦χ, χ↦-ψ, and ε↦ε-ψχ. A short verification that this map preserves the field equations or the potential-space metric would make the claim self-contained.
  2. [Application to Weyl seeds] The application to harmonic scalar-vacuum Weyl seeds with independent gravitational and scalar harmonics is illustrated by generating charged branches via frozen-ModMax Harrison maps and gravito-rotational branches via Ehlers. Adding one explicit numerical or functional example of the final quadratures for k, ω, and A_φ would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments help clarify the scope of our results on EMMSF symmetries and strengthen the connection between the Hamiltonian and Lie-algebraic structures. We address each point below and will incorporate revisions as indicated.

read point-by-point responses

  1. Referee: [Abstract and frozen-regime discussion] Abstract and the paragraph on the frozen EMMSF regime: the central claim states that the symmetries are completed for both EMSF and EMMSF, yet the ModMax sectorial deformations and the coexistence of electric and magnetic Harrison transformations are explicitly restricted to the frozen regime (v=v0, w=w0) with the conditions dw=0 and d[(v²+w²)/w]=0 imposed. This limitation is load-bearing for the EMMSF portion of the title and abstract; the manuscript should state upfront whether the potential formulation and Lie-algebra actions extend to generic (non-frozen) ModMax constitutive relations or whether additional obstructions arise.

    Authors: We agree that the EMMSF results are developed specifically in the frozen regime. The manuscript already derives that coexistence of electric and magnetic Harrison transformations imposes dw=0 and d[(v²+w²)/w]=0, which selects precisely the frozen sector (v=v0, w=w0). For generic (non-frozen) ModMax constitutive relations, these conditions are not satisfied and the sectorial Lie-algebra actions do not close in the same way; additional obstructions arise from the nonlinear electromagnetic constitutive relations. We will revise the abstract and the opening paragraph of the frozen-regime discussion to state this limitation explicitly upfront, making clear that the completed symmetries for EMMSF refer to the frozen case. revision: yes

  2. Referee: [Hamiltonian formulation and Noether charges] Section on Hamiltonian formulation and Noether charges: the statement that the affine geodesic energy is constant in harmonic branches of the A,B,C one-forms is used to control the quadrature for k, with ω and A_φ obtained from Noether charges along the Killing directions of ε and χ. It is not shown how these charges relate to the Casimir invariants of the sectorial algebras; an explicit relation or example would confirm that the Hamiltonian approach is consistent with the Lie-algebra structure derived earlier.

    Authors: The Casimir invariants of the sectorial algebras are the conserved quantities generated by the Killing vectors in the potential space; in the harmonic branches these coincide with the Noether charges associated with the symmetries of ε and χ. The constancy of the affine geodesic energy is itself a consequence of the Casimir being preserved along the flow. We will add a short explicit paragraph (or a brief example in the Ehlers or Harrison sector) in the revised manuscript that directly identifies the Noether charges with the relevant Casimir invariants, confirming consistency between the Hamiltonian and Lie-algebraic descriptions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations build on independent Ernst formalism

full rationale

The paper derives visible symmetries and their solvable Lie algebra directly from the real potential space coordinates (f,ε,ψ,χ,κ) after assuming the field equations admit an Ernst-like formulation. Hidden symmetries (Ehlers, Harrison) are characterized on invariant subspaces using standard Lie-group actions. The frozen-regime restriction for EMMSF is obtained as a derived condition from requiring coexistence of electric and magnetic Harrison transformations (imposing dw=0 and d[(v²+w²)/w]=0), not presupposed by definition. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations appear; the work extends classic Ehlers-Harrison results from independent literature without circular closure.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the existence of an Ernst-like potential formulation for the EMSF and EMMSF actions, the stationarity and axisymmetry assumptions that reduce the system to a finite-dimensional potential space, and the standard Lie-algebra methods for identifying isometries of that space. No new free parameters or invented entities are introduced; the work uses the conventional Einstein equations with Maxwell or ModMax matter and a massless scalar field.

axioms (2)
  • domain assumption The spacetime admits two commuting Killing vectors (stationary and axisymmetric) allowing reduction to Ernst-like potentials.
    Invoked throughout the abstract when defining the real potential space (f,ε,ψ,χ,κ).
  • standard math The field equations derive from the Einstein-Maxwell-Scalar or Einstein-ModMax-Scalar action.
    Standard background assumption of the theories under study.

pith-pipeline@v0.9.0 · 5919 in / 1524 out tokens · 35414 ms · 2026-05-20T10:13:08.855408+00:00 · methodology

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Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages · 14 internal anchors

  1. [1]

    Visible charges QKϵ =p ϵ = ˙ϵ−ψ˙χ 2f2 , QKχ =p χ =− ψ(˙ϵ−ψ˙χ) 2f2 − ˙χ 2f κ2 , QKψ =p ψ +χp ϵ =− κ2 ˙ψ 2f + χ(˙ϵ−ψ˙χ) 2f2 , QDg =f p f +ϵp ϵ + 1 2 ψpψ + 1 2 χpχ = ˙f 2f + ϵ(˙ϵ−ψ˙χ) 2f2 − ψκ2 ˙ψ 4f − χψ(˙ϵ−ψ˙χ) 4f2 − χ˙χ 4f κ2 , QDs =κp κ +χp χ −ψp ψ = ˙κ βκ − χψ(˙ϵ−ψ˙χ) 2f2 − χ˙χ 2f κ2 + ψκ2 ˙ψ 2f . (43)

    work page

  2. [2]

    (44) The Casimir is Cgrav =Q 2 D − QT QE =f 2(p2 f +p 2 ϵ) = ˙f2 + ˙ϵ2 4f2

    Ehlers gravitational scenario The Ehlers charges are QT =p ϵ = ˙ϵ 2f2 , QD =f p f +ϵp ϵ = ˙f 2f + ϵ˙ϵ 2f2 , QE = 2ϵf pf + (ϵ2 −f 2)pϵ = ϵ ˙f f + (ϵ2 −f 2)˙ϵ 2f2 . (44) The Casimir is Cgrav =Q 2 D − QT QE =f 2(p2 f +p 2 ϵ) = ˙f2 + ˙ϵ2 4f2 . (45)

    work page

  3. [3]

    (46) The Casimir is CM =Q 2 JM −4γQ PM QHM =f 2p2 f −4γf κ 2p2 χ +β 2κ2p2 κ + 2βf κpf pκ = ˙f2 4f2 − γ˙χ2 f κ2 + ˙κ2 κ2 + ˙f˙κ f κ

    Magnetic Harrison scenario The magnetic Harrison charges are QPM =p χ =− ˙χ 2f κ2 , QJM =f p f + 2γχpχ +βκp κ = ˙f 2f − γχ˙χ f κ2 + ˙κ κ , QHM =f χp f + (f κ2 +γχ 2)pχ +βκχp κ = χ ˙f 2f − ˙χ 2 − γχ2 ˙χ 2f κ2 + χ˙κ κ . (46) The Casimir is CM =Q 2 JM −4γQ PM QHM =f 2p2 f −4γf κ 2p2 χ +β 2κ2p2 κ + 2βf κpf pκ = ˙f2 4f2 − γ˙χ2 f κ2 + ˙κ2 κ2 + ˙f˙κ f κ. (47)

    work page

  4. [4]

    (48) The Casimir is CE =Q 2 JE −4γQ PE QHE =f 2p2 f − 4γf κ2 p2 ψ +β 2κ2p2 κ −2βf κp f pκ = ˙f2 4f2 − γκ2 ˙ψ2 f + ˙κ2 κ2 − ˙f˙κ f κ

    Electric Harrison scenario The electric Harrison charges are QPE =p ψ =− κ2 ˙ψ 2f , QJE =f p f + 2γψpψ −βκp κ = ˙f 2f − γκ2ψ ˙ψ f − ˙κ κ , QHE =f ψp f + f κ2 +γψ 2 pψ −βκψp κ = ψ ˙f 2f − ˙ψ 2 − γκ2ψ2 ˙ψ 2f − ψ˙κ κ . (48) The Casimir is CE =Q 2 JE −4γQ PE QHE =f 2p2 f − 4γf κ2 p2 ψ +β 2κ2p2 κ −2βf κp f pκ = ˙f2 4f2 − γκ2 ˙ψ2 f + ˙κ2 κ2 − ˙f˙κ f κ. (49)

    work page

  5. [5]

    Ehlers gravitational scenario:Since the electro- magnetic sector is switched off in this scenario, the for- mulas (44) and (45) remain the same

    ModMax Harrison scenario a. Ehlers gravitational scenario:Since the electro- magnetic sector is switched off in this scenario, the for- mulas (44) and (45) remain the same. In the frozen ModMax branch, where v and w are con- stants, the Harrison algebras keep the same commutation relations as in the Maxwell case. However, the electro- magnetic charges mus...

    work page

  6. [6]

    In the harmonic branch, k,ζ = K0 ρs 2 ,ζ, where ζ = ρ + iz

    and in (42) of Appendix A in [ 2]. In the harmonic branch, k,ζ = K0 ρs 2 ,ζ, where ζ = ρ + iz. If R is defined byR ,ζ =ρs 2 ,ζ,then k=k 0 +K 0 R.(57) Thus K0 is not an arbitrary integration constant detached from the target geometry, it is the geodesic energy of the harmonic curve. E. Visible quadratures Let X = X A(Y )∂A be a Killing vector of the target...

    work page

  7. [7]

    Since A0 = ¯A0 and B0 = 0, the A-equation reduces to ρ−1D(ρA0) = 0 , which follows from the harmonicity of sf

    and (20) in [ 2] consequently. Since A0 = ¯A0 and B0 = 0, the A-equation reduces to ρ−1D(ρA0) = 0 , which follows from the harmonicity of sf. Similarly, the C-equation reduces to ρ−1D(ρC0) = 0, which is a direct consequence of the harmonicity of sκ. Consequently, the two-harmonic seed constitutes an exact scalar-vacuum seed of the generalized Ernst system...

    work page 2025

  8. [8]

    Ple- ban´ ski–Demia´ nski ` a la Ehlers–Harrison: exact rotat- ing and accelerating type I black holes

    Barrientos, J., Cisterna, A., Pallikaris, K., 2024. Ple- ban´ ski–Demia´ nski ` a la Ehlers–Harrison: exact rotat- ing and accelerating type I black holes. Gen. Rel. Grav. 56, 111. doi:doi:10.1007/s10714-024-03304-x, arXiv:2309.13656

    work page doi:10.1007/s10714-024-03304-x 2024

  9. [9]

    Generalized Einstein- ModMax-ScalarField theories and new exact solutions arXiv:2603.26073

    Bixano, L., Matos, T., 2026a. Generalized Einstein- ModMax-ScalarField theories and new exact solutions arXiv:2603.26073

    work page arXiv

  10. [10]

    Generalized Ernst Poten- tials for arbitrary Dilatonic TheoriesarXiv:2603.02384

    Bixano, L., Matos, T., 2026b. Generalized Ernst Poten- tials for arbitrary Dilatonic TheoriesarXiv:2603.02384

    work page arXiv

  11. [11]

    Generalised Harrison transformations and black diholes in Einstein-ModMax

    Bokuli´ c, A., Herdeiro, C.A.R., 2025. Generalised Harrison transformations and black diholes in Einstein-ModMax. JHEP 10, 091. doi:doi:10.1007/JHEP10(2025)091, arXiv:2507.16926

    work page doi:10.1007/jhep10(2025)091 2025

  12. [12]

    New formulation of the axially symmet- ric gravitational field problem

    Ernst, F.J., 1968a. New formulation of the axially symmet- ric gravitational field problem. Phys. Rev. 167, 1175–1179. doi:doi:10.1103/PhysRev.167.1175

    work page doi:10.1103/physrev.167.1175

  13. [13]

    New Formulation of the Axially Sym- metric Gravitational Field Problem

    Ernst, F.J., 1968b. New Formulation of the Axially Sym- metric Gravitational Field Problem. II. Phys. Rev. 168, 1415–1417. doi:doi:10.1103/PhysRev.168.1415

    work page doi:10.1103/physrev.168.1415

  14. [14]

    Integrable Systems in Stringy Gravity

    Gal’tsov, D.V., 1995. Integrable systems in stringy gravity. Phys. Rev. Lett. 74, 2863–2866. doi:doi: 17 10.1103/PhysRevLett.74.2863,arXiv:hep-th/9410217

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.74.2863 1995

  15. [15]

    Symmetries of the stationary Einstein--Maxwell--dilaton theory

    Galtsov, D.V., Garcia, A.A., Kechkin, O.V., 1995. Sym- metries of the stationary Einstein-Maxwell dilaton theory. Class. Quant. Grav. 12, 2887–2903. doi:doi:10.1088/0264- 9381/12/12/007,arXiv:hep-th/9504155

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/0264- 1995

  16. [16]

    Ehlers--Harrison--Type Transformations in Dilaton--Axion Gravity

    Galtsov, D.V., Kechkin, O.V., 1994. Ehlers-Harrison type transformations in dilaton - axion gravity. Phys. Rev. D 50, 7394–7399. doi:doi:10.1103/PhysRevD.50.7394, arXiv:hep-th/9407155

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.50.7394 1994

  17. [17]

    Hidden Symmerties in Dilaton--Axion Gravity

    Gal’tsov, D.V., Kechkin, O.V., 1995. Hidden symme- tries in dilaton - axion gravity, in: International Work- shop on Geometry and Integrable Models, pp. 78–95. arXiv:gr-qc/9606014

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  18. [18]

    Ehlers-Harrison transformations and black holes in Dilaton-Axion Gravity with multiple vector fields

    Gal’tsov, D.V., Letelier, P.S., 1997. Ehlers-Harrison trans- formations and black holes in dilaton - axion gravity with multiple vector fields. Phys. Rev. D 55, 3580–3592. doi:doi: 10.1103/PhysRevD.55.3580,arXiv:gr-qc/9612007

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.55.3580 1997

  19. [19]

    New Solutions of the Einstein- Maxwell Equations from Old

    Harrison, B.K., 1968. New Solutions of the Einstein- Maxwell Equations from Old. J. Math. Phys. 9, 1744. doi:doi:10.1063/1.1664508

    work page doi:10.1063/1.1664508 1968

  20. [20]

    Black holes in scalar multi- polar universes

    Herdeiro, C.A.R., 2025. Black holes in scalar multi- polar universes. Phys. Lett. B 860, 139160. doi:doi: 10.1016/j.physletb.2024.139160,arXiv:2410.12950

    work page doi:10.1016/j.physletb.2024.139160 2025

  21. [21]

    Charging Symmetries and Linearizing Potentials for Einstein-Maxwell Dilaton-Axion Theory

    Herrera-Aguilar, A., Kechkin, O., 1998. Charging symmetries and linearizing potentials for Einstein- Maxwell dilaton - axion theory. Mod. Phys. Lett. A 13, 1907–1914. doi:doi:10.1142/S0217732398002011, arXiv:hep-th/9806247

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1142/s0217732398002011 1998

  22. [22]

    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. doi:doi:10.1063/1.523458

    work page doi:10.1063/1.523458 1977

  23. [23]

    Symmetries of the Stationary Einstein-Maxwell Field Equations

    Kinnersley, W., Chitre, D.M., 1977. Symmetries of the Stationary Einstein-Maxwell Field Equations. 2. J. Math. Phys. 18, 1538–1542. doi:doi:10.1063/1.523459

    work page doi:10.1063/1.523459 1977

  24. [24]

    Symmetries of the Stationary Einstein-Maxwell Field Equations

    Kinnersley, W., Chitre, D.M., 1978. Symmetries of the Stationary Einstein-Maxwell Field Equations. 3. J. Math. Phys. 19, 1926–1931. doi:doi:10.1063/1.523912

    work page doi:10.1063/1.523912 1978

  25. [25]

    Lie Groups Beyond an Introduction

    Knapp, A.W., 1996. Lie Groups Beyond an Introduction. volume 140 ofProgress in Mathematics. 1 ed., Birkh¨ auser Boston, Boston, MA. doi:doi:10.1007/978-1-4757-2453-0

    work page doi:10.1007/978-1-4757-2453-0 1996

  26. [26]

    Metric of a rotating charged magnetized sphere

    Manko, V.S., Mej´ ıa, I.M., Ruiz, E., 2020. Metric of a rotating charged magnetized sphere. Phys. Lett. B 803, 135286. doi:doi:10.1016/j.physletb.2020.135286, arXiv:1912.08884

    work page doi:10.1016/j.physletb.2020.135286 2020

  27. [27]

    The linear problem for the five- dimensional projective field theory

    Matos, T., 1986. The linear problem for the five- dimensional projective field theory. Astron. Nachr. 307, 317–320. doi:doi:10.1002/asna.2113070521

    work page doi:10.1002/asna.2113070521 1986

  28. [28]

    Sl(3, r) representation for invariance transformations in five-dimensional gravity

    Matos, T., 1988. Sl(3, r) representation for invariance transformations in five-dimensional gravity. Physics Let- ters A 131, 423–426. doi:doi:https://doi.org/10.1016/0375- 9601(88)90292-7

    work page doi:10.1016/0375- 1988

  29. [29]

    5D Axisymmetric Stationary Solutions as Harmonic Maps

    Matos, T., 1994. 5-D axisymmetric stationary solutions as harmonic maps. J. Math. Phys. 35, 1302–1321. doi:doi: 10.1063/1.530590,arXiv:gr-qc/9401009

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1063/1.530590 1994

  30. [30]

    Class of Einstein-Maxwell Phantom Fields: Rotating and Magnetised Wormholes

    Matos, T., 2010. Class of Einstein-Maxwell Phantom Fields: Rotating and Magnetised Wormholes. Gen. Rel. Grav. 42, 1969–1990. doi:doi:10.1007/s10714-010-0976-6, arXiv:0902.4439

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s10714-010-0976-6 2010

  31. [31]

    Class of Einstein-Maxwell-Dilaton-Axion Space-Times

    Matos, T., Miranda, G., Sanchez-Sanchez, R., Wieder- hold, P., 2009. Class of Einstein-Maxwell-Dilaton- Axion Space-Times. Phys. Rev. D 79, 124016. doi:doi: 10.1103/PhysRevD.79.124016,arXiv:0905.4097

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.79.124016 2009

  32. [32]

    Stationary Dilatons with Arbitrary Electromagnetic Field

    Matos, T., Mora, C., 1997. Stationary dilatons with arbitrary electromagnetic field. Class. Quant. Grav. 14, 2331–2340. doi:doi:10.1088/0264-9381/14/8/027, arXiv:hep-th/9610013

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/0264-9381/14/8/027 1997

  33. [33]

    Rotating 5D-Kaluza-Klein Space-Times from Invariant Transformations

    Matos, T., Nunez, D., Estevez, G., Rios, M., 2000a. Rotating 5-D Kaluza-Klein space-times from invariant transformations. Gen. Rel. Grav. 32, 1499–1525. doi:doi: 10.1023/A:1001982001694,arXiv:gr-qc/0001039

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1023/a:1001982001694

  34. [34]

    Class of Einstein--Maxwell Dilatons

    Matos, T., Nunez, D., Quevedo, H., 1995. Class of Einstein-Maxwell dilatons. Phys. Rev. D 51, R310–R313. doi:doi:10.1103/PhysRevD.51.R310, arXiv:gr-qc/9510042

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.51.r310 1995

  35. [35]

    Class of Eistein-Maxwell Dilatons

    Matos, T., Nunez, D., Rios, M., 2000b. Class of Einstein- Maxwell dilatons, an ansatz for new families of rotating solutions. Class. Quant. Grav. 17, 3917–3934. doi:doi: 10.1088/0264-9381/17/18/323,arXiv:gr-qc/0008068

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/0264-9381/17/18/323

  36. [36]

    Axisymmetric Stationary Solutions as Harmonic Maps

    Matos, T., Plebanski, J., 1994. Axisymmetric stationary solutions as harmonic maps. Gen. Rel. Grav. 26, 477. doi:doi:10.1007/BF02108050,arXiv:gr-qc/9402044

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/bf02108050 1994

  37. [37]

    The quadratures are defined in equation (7) of Ref

    Note1, . The quadratures are defined in equation (7) of Ref. [2]

    work page