Pith. sign in

REVIEW 3 major objections 4 minor 40 references

For the Standard Model extended by one or two real gauge-singlet scalars, all scalar Lie point symmetries — strict variational, divergence, and non-variational — are classified into a short list of inequivalent algebras, with algorithms tha

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-02 21:07 UTC pith:X6LVXQXC

load-bearing objection Careful and useful classification, held back by an unproved technical step and hidden computer algebra; worth reviewing, not rejecting. the 3 major comments →

arxiv 2602.21122 v2 pith:X6LVXQXC submitted 2026-02-24 hep-ph hep-thmath-phmath.MP

Scalar Lie point symmetries of the Standard Model with one or two real gauge singlets

M. Aa. Solberg This is my paper
classification hep-ph hep-thmath-phmath.MP
keywords scalar Lie point symmetriesStandard Model singlet extensionsvariational symmetriesdivergence symmetriesnon-variational symmetriessymmetry classificationHiggs portal
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper aims to fully classify the scalar Lie point symmetries — continuous symmetries of the scalar field equations — of the Standard Model extended by one or two real gauge-singlet scalars (SM+S and SM+2S). It claims that, up to affine reparametrization, only four inequivalent symmetry algebras occur for SM+S and eleven for SM+2S, and it separates them into strict variational, divergence, and non-variational types. It also provides a direct parameter-based algorithm that, for any numerical potential, identifies the maximal symmetry algebra of each type without solving the determining equations. A complete classification matters because variational symmetries give conserved currents and constrain model parameters, while non-variational symmetries (such as the free-singlet scaling) are symmetries of the equations of motion only. If correct, the paper gives model-builders a complete catalogue of the continuous scalar symmetries available to singlet-extension models.

Core claim

The paper establishes that the scalar Lie point symmetries of the SM+S and SM+2S are completely classified. For the SM+S, the realizable, inequivalent Euler–Lagrange symmetry algebras are exactly a(1)⊕u(1)_Y (a free, massless singlet), s_h⊕u(1)_Y (massless singlet with a linear term), s_c⊕u(1)_Y (free massive singlet), and u(1)_Y (the hypercharge rephasing, always present). For the SM+2S, the list grows to 11 inequivalent Euler–Lagrange algebras, including so(2) rotations corresponding to a complex singlet with a global U(1), the affine algebra a(2) of the free two-singlet kinetic theory, and mixed shift/scaling algebras, with the variational and strict variational subalgebras separately enu

What carries the argument

The load-bearing machinery is Corollary 1, which characterizes the three symmetry types for Lagrangians of the form L = T − V. For a scalar point symmetry X = η^i(φ) ∂_{φ^i} with constant part a^i and linear-potential coefficients α_i, X is a strict variational symmetry if and only if pr X(T)=0 and a^i α_i = 0; a divergence symmetry if pr X(T) is a total divergence and (a^i α_i ≠ 0 or the divergence is nonzero); and non-variational if E(pr X(T)) ≠ 0. The proof rests on the identity pr X(E(L)) = E(pr X(L)) − D*_Q E(L) and on the hypothesis that E(L)=0 has no polynomial consequences, which forces pr X(V) to be a constant. Affine reparametrizations (orthogonal changes of singlet basis and const

Load-bearing premise

The classification of strict variational, divergence, and non-variational symmetries assumes that the SM+S and SM+2S field equations imply no nonzero polynomial relation among the scalar fields, a hypothesis the paper invokes but never verifies.

What would settle it

Pick a concrete parameter point, e.g. the SM+S free massive singlet (µ_s² ≠ 0, all other singlet couplings zero), write out the five scalar Euler–Lagrange equations, and test whether the system implies any nonzero polynomial identity p(ϕ_1,...,ϕ_5)=0. If such an identity exists, Theorem 1's conclusion that pr X(V) is constant fails, and the s_c⊕u(1)_Y classification of symmetry types would need revision.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • For any numerical SM+S or SM+2S potential, the symmetry algebras g_EL, g_var, and g_svar can be read off from parameter conditions alone, making systematic parameter scans feasible without solving determining equations.
  • The theorems give a general criterion for when a scalar field symmetry is variational (and hence yields a Noether current) versus merely a symmetry of the field equations, applicable beyond these two models to any Lagrangian with a potential of the assumed form.
  • The variational symmetries of SM+S reduce to the singlet shift (strict when α=0, divergence otherwise) plus hypercharge; in SM+2S the only variational generators are the two singlet shifts, the rotation s_2∂_{s1} − s_1∂_{s2}, and hypercharge, which tells model-builders which continuous symmetries can be imposed consistently.
  • Models realizing the a(2)⊕u(1)_Y algebra correspond to a free, massless two-singlet sector; models realizing so(2)⊕u(1)_Y are equivalent to a complex singlet with a global U(1), matching known pseudo-Goldstone dark matter setups.

Where Pith is reading between the lines

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

  • The 'no polynomial consequences' hypothesis is the main unverified link; checking it explicitly for representative parameter points via elimination of derivatives (e.g., by computing a Gröbner basis of the differential ideal of the field equations) would confirm the classification's scope.
  • The parameter-based algorithms could be turned into automated scans that flag which symmetry algebra a given benchmark point in the singlet-extension parameter space realizes, useful for dark matter and baryogenesis studies.
  • The classification's approach — start from the symmetry algebra of the kinetic terms and reduce by reparametrizations — suggests a route to SM+KS with K>2, though the reduction tree complexity grows rapidly; the paper explicitly notes the boost algebra k⊕u(1)_Y with eigenvalues ±1 is not realizable in SM+2S, a caution for naive symmetry searches.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper classifies scalar Lie point symmetries for the Standard Model extended by one or two real gauge-singlet scalars (SM+S and SM+2S), distinguishing strict variational, divergence, and non-variational (Euler–Lagrange) symmetries. For SM+S it finds four inequivalent realizable algebras; for SM+2S eleven. It provides parameter-based algorithms to determine the symmetry algebra of any numerical instance without solving determining equations, and it proves general results (Theorem 1, Proposition 1, Corollary 1, Proposition 2) characterizing symmetry types for Lagrangians with potentials. The classification is obtained by solving determining equations with the Mathematica packages SYM/Reduce over a reduction tree of 31 parameter branches.

Significance. If the results hold, the paper delivers a useful catalogue of scalar Lie point symmetries for two popular BSM models and an efficient algorithm for symmetry identification in parameter scans. The general characterization in Corollary 1 and the explicit proof of Proposition 3 are valuable extensions of earlier work. The g_EL classification is derived directly from the determining equations and is supported by explicit examples, e.g., the Leaf 6 analysis and the calculations around Eqs. (4.113)–(4.115). However, the SVS/DS/NVS classification rests on a structural hypothesis that is stated but never verified for the physical models, which limits confidence in the variational and strict-variational lists.

major comments (3)
  1. [§2.4 (Theorem 1, Corollary 1) and §§3.3, 4.5] The theorem and corollary assume that the system E(L)=0 'has no polynomial consequences'. This hypothesis is never verified for SM+S or SM+2S. It is essential for the step concluding that prX(V) is constant from on-shell vanishing of its partial derivatives, and thereby underpins the SVS/DS/NVS classification of every generator. If some branch of the reduction tree admitted a nonzero polynomial relation among the scalar fields, a generator classified as non-variational could actually be a divergence symmetry (or vice versa), changing the g_var and g_svar lists in Eqs. (5.4)–(5.5). The authors should either prove the property for these field equations (e.g., by a local-solvability/Cauchy–Kovalevskaya argument using the nondegenerate kinetic terms) or provide a direct per-leaf computation of prX(L) and E(prX(L)) for each generator in the set ℵ, thereby avoiding the hypothesis altogether.
  2. [§4.4 and Appendix A] The exhaustiveness of the computational classification is not fully transparent. The text states that SYM/Reduce solves the determining equations for each of the 31 leaves, but the solving steps and intermediate differential-elimination results are not shown. To support the claimed completeness of the reduction tree and the resulting classification, the authors should make the code available or include a detailed appendix (or supplementary material) with, for each leaf, the reduced determining equations and the final solutions, together with a justification that the branching in Figures 1–3 covers all possible parameter values.
  3. [Eq. (5.3)] The summary list contains 'sc_2⊕u(1)_Y' as one of the eleven inequivalent algebras, but Section 4.5.1 shows that sc_1≂O sc_2≂O sc, so sc_2⊕u(1)_Y is not inequivalent to sc⊕u(1)_Y, which is already listed. The intended entry is presumably 'sc_1⊕sc_2⊕u(1)_Y', which appears as one of the three (2+1)d algebras in (4.157). As written, the list contradicts the 'inequivalent' claim and the count of eleven; please correct this.
minor comments (4)
  1. [Notation in §5 and §4.5] The notation 'sc_2' is ambiguous: it is used for a one-dimensional algebra acting on the second singlet, but in (5.3) it appears to represent a two-dimensional direct sum. Recommend writing 'sc_1⊕sc_2' explicitly when that is intended.
  2. [Figure 2] The reduction tree is very dense. A legend distinguishing red leaves, intermediate nodes, and '∅' nodes, and perhaps a higher-resolution or simplified version, would improve readability.
  3. [§2.4, proof of Proposition 1] The 'no cancellations' argument between Eqs. (2.55) and (2.56) is somewhat terse. A sentence explaining why, under the assumptions on T, E_i(prX(T)) cannot contain a pure polynomial term in the φ variables would help the reader follow the proof.
  4. [References] Several standard results are cited to [19], the author's previous work. Since the present paper aims to be self-contained, consider restating the most important definitions (e.g., the equivalence notation ≂O and the affine-vector-field notation) with equation numbers, rather than relying on the earlier reference.

Circularity Check

0 steps flagged

No significant circularity: the symmetry algebras are derived from determining equations and parameter reductions, not from the quantities they are claimed to classify.

full rationale

The paper's central claim is a classification of scalar Lie point symmetries obtained by solving the linearized symmetry condition (2.18) for the Euler-Lagrange equations and then organizing the solutions by orthogonal affine reparametrizations. The parameter conditions such as (3.21)-(3.26) and (4.69)-(4.71) are outputs of the determining-equation solve, not fitted inputs; the classification is not defined in terms of the listed algebras. Theorems 1 and Corollary 1 are proven in the text, albeit under the explicit hypothesis that E(L)=0 'has no polynomial consequences' (Section 2.4), and this hypothesis is not verified for SM+S/SM+2S in Sections 3.3 and 4.5. That is an unverified assumption (a correctness risk) rather than a circular step: the characterization would be invalid if the hypothesis failed, but it does not reduce the claimed prediction to the theorem's input by construction. Self-citations to [19] and [26] are contextual (overlapping review, standard reparametrization facts) and are not load-bearing; the main theorem used later is proved in this paper. No fitted-input-called-prediction, self-definitional, or renaming pattern is present, so a low circularity score is appropriate.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The paper fits no numerical constants and introduces no new particles, forces, or fields. The only hand-chosen quantities are computational reparametrization parameters γ, θ and a scan tolerance t, which do not enter the classification. The main undeclared input is the correctness of the computer algebra and the unproven no-polynomial-consequences property.

axioms (5)
  • domain assumption The system E(L)=0 for SM+S and SM+2S has no polynomial consequences: no nonzero polynomial p(φ)=0 follows from the field equations.
    Theorem 1 and Corollary 1 use this to upgrade E(prX(V))=0 from on-shell to identically zero. The paper never proves this for the singlet-extended Standard Model systems it classifies.
  • domain assumption All terms in the kinetic sector T are either at least quadratic in the gauge fields or at least linear in field derivatives.
    This assumption in Theorem 1 ensures prX(T) cannot be a nonzero constant. It is stated as being chosen to fit NHDM+KS models, and the SM+S/SM+2S kinetic terms satisfy it, but no detailed verification is given.
  • ad hoc to paper The Mathematica packages SYM and Reduce correctly solve the determining equations for all 31 leaves.
    The classification depends on computer algebra outputs, but no scripts, notebooks, or logs are provided. Errors in a leaf would change the classification.
  • standard math Orthogonal affine reparametrizations, together with rescaling of the group parameter, are the correct equivalence relation for identifying inequivalent symmetry realizations.
    Proposition 2 proves preservation of symmetry type under affine reparametrizations; the restriction to orthogonal linear part preserves canonical kinetic terms. This is standard but is used as the definition of equivalence.
  • domain assumption Renormalization preserves variational symmetries in the absence of anomalies, so tree-level symmetry analysis is relevant for the effective potential.
    Used in Section 4.2.1 as motivation for analyzing the tree-level potential. It does not affect the mathematical classification itself.

pith-pipeline@v1.3.0-alltime-deepseek · 34624 in / 14661 out tokens · 149539 ms · 2026-08-02T21:07:12.961021+00:00 · methodology

0 comments
read the original abstract

We present a classification of all scalar Lie point symmetries of the Standard Model with one or two real gauge-singlet scalars (SM+S and SM+2S). By analyzing the associated field equations, we identify all realizable and inequivalent Lie point symmetry algebras of these models, distinguishing strict variational, variational (including divergence symmetries), and Euler--Lagrange cases. In addition, we devise efficient algorithms that, for any given numerical instance of the models, determine the Lie point symmetry algebra in each of the three categories by a parameter-based decision procedure using affine reparametrizations and simple parameter tests, thereby avoiding explicit symmetry analysis and the need to derive and solve the determining equations. Finally, we prove several relevant general results, including a characterization of the three disjoint types of Lie point symmetry generators -- strict variational, divergence, and non-variational -- for a broad class of Lagrangians with potentials, including the SM+S and SM+2S.

Figures

Figures reproduced from arXiv: 2602.21122 by M. Aa. Solberg.

Figure 1
Figure 1. Figure 1: Reduction tree, Branches I and II. Each path starting from the root at the top and terminating at a leaf (marked in red) at the bottom corresponds to a reduced potential. For each leaf, the determining equations are solved and the corresponding symmetries are derived. Leaf 1 is discussed in Section 4.4.1, while Leaf 2 is treated in Section 4.4.2. 4.4.2 Branch II: λ1111 = 0, λ1122 ≠ 0 Now consider the case … view at source ↗
Figure 2
Figure 2. Figure 2: Reduction tree, Branches III and IV. Red boxes indicate key nodes, numbered sequentially (3–31). All but one of these are leaves corresponding to reduced potentials, while Node 3 represents an intermediate reduction stage. Some leaves may still allow further (possibly nonlinear) reductions. For each key node, the determining equations are solved and the corresponding symmetries are derived. The empty set ∅… view at source ↗
Figure 3
Figure 3. Figure 3: Reduction tree, Branch IV (continued). Each path starting from the root in [PITH_FULL_IMAGE:figures/full_fig_p033_3.png] view at source ↗

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

40 extracted references · 10 linked inside Pith

  1. [1]

    Silveira and A

    V. Silveira and A. Zee,Scalar phantoms,Physics Letters B161(1985) 136–140

    1985

  2. [2]

    McDonald,Gauge singlet scalars as cold dark matter.,Physical review

    J. McDonald,Gauge singlet scalars as cold dark matter.,Physical review. D, Particles and fields50 6(1994) 3637–3649

    1994

  3. [3]

    Burgess, M

    C. Burgess, M. Pospelov and T. ter Veldhuis,The minimal model of nonbaryonic dark matter: a singlet scalar,Nuclear Physics B619(Dec., 2001) 709–728

    2001

  4. [4]

    Arcadi, A

    G. Arcadi, A. Djouadi and M. Raidal,Dark matter through the higgs portal,Physics Reports842(2020) 1–180

    2020

  5. [5]

    Ahriche,What is the criterion for a strong first order electroweak phase transition in singlet models?,Physical Review D75(Apr., 2007)

    A. Ahriche,What is the criterion for a strong first order electroweak phase transition in singlet models?,Physical Review D75(Apr., 2007)

    2007

  6. [6]

    J. R. Espinosa, T. Konstandin and F. Riva,Strong Electroweak Phase Transitions in the Standard Model with a Singlet,Nucl. Phys. B854(2012) 592–630, [1107.5441]. 48

    Pith/arXiv arXiv 2012

  7. [7]

    Ellis, M

    J. Ellis, M. Lewicki, M. Merchand, J. M. No and M. Zych,The scalar singlet extension of the Standard Model: gravitational waves versus baryogenesis,JHEP01 (2023) 093, [2210.16305]

    Pith/arXiv arXiv 2023

  8. [8]

    Yu, T.-P

    Y. Yu, T.-P. Tang and L. Feng,New constraints on singlet scalar dark matter model with lz, invisible higgs decay and gamma-ray line observations, 2025

    2025

  9. [9]

    Lewis and M

    I. Lewis and M. Sullivan,Benchmarks for double higgs production in the singlet-extended standard model at the lhc,Physical Review D96(Aug., 2017)

    2017

  10. [10]

    C.-Y. Chen, J. Kozaczuk and I. M. Lewis,Non-resonant collider signatures of a singlet-driven electroweak phase transition,Journal of High Energy Physics2017 (Aug., 2017)

    2017

  11. [11]

    Tofighi, O

    A. Tofighi, O. Ghodsi and M. Saeedhoseini,Phase transition in multi-scalar-singlet extensions of the standard model,Physics Letters B748(Sept., 2015) 208–211

    2015

  12. [12]

    V. R. Shajiee and A. Tofighi,Electroweak phase transition, gravitational waves and dark matter in two scalar singlet extension of the standard model,The European Physical Journal C79(Apr., 2019)

    2019

  13. [13]

    Barger, P

    V. Barger, P. Langacker, M. McCaskey, M. Ramsey-Musolf and G. Shaughnessy, Complex singlet extension of the standard model,Phys. Rev. D79(Jan, 2009) 015018

    2009

  14. [14]

    Kannike and M

    K. Kannike and M. Raidal,Phase Transitions and Gravitational Wave Tests of Pseudo-Goldstone Dark Matter in the Softly Broken U(1) Scalar Singlet Model, Phys. Rev. D99(2019) 115010, [1901.03333]

    Pith/arXiv arXiv 2019

  15. [15]

    Arina, A

    C. Arina, A. Beniwal, C. Degrande, J. Heisig and A. Scaffidi,Global fit of pseudo-nambu-goldstone dark matter,Journal of High Energy Physics2020(Apr.,

  16. [16]

    Symmetries in quantum field theory

    D. B. Skinner, “Symmetries in quantum field theory.” Chapter 6 of the lecture notes forQuantum Field Theory II(University of Cambridge, Part III Mathematical Tripos), 2015

    2015

  17. [17]

    Alvarez-Gaum´ e and F

    L. Alvarez-Gaum´ e and F. Zamora,Duality in quantum field theory (and string theory), Sept., 1997

    1997

  18. [18]

    Lie and F

    S. Lie and F. Engel,Theorie der Transformationsgruppen, vol. 1-3. B.G. Teubner, Leipzig, 1888-1893

  19. [19]

    M. A. Solberg,Lie symmetry analysis of the two-higgs-doublet model field equations, Physica Scripta101(jan, 2026) 035209

    2026

  20. [20]

    P. J. Olver,Applications of Lie Groups to Differential Equations, vol. 107 of Graduate Texts in Mathematics. Springer, New York, 2 ed., 1993, 10.1007/978-1-4612-4350-2

    doi:10.1007/978-1-4612-4350-2 1993

  21. [21]

    Lectures on Lie Groups and Differential Equations

    P. J. Olver, “Lectures on Lie Groups and Differential Equations.” https://www-users.cse.umn.edu/~olver/sm.html, 2012. 49

    2012

  22. [22]

    P. E. Hydon,Symmetry Methods for Differential Equations: A Beginner’s Guide, vol. 22 ofCambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2000

    2000

  23. [23]

    G. W. Bluman, A. F. Cheviakov and S. C. Anco,Applications of Symmetry Methods to Partial Differential Equations, vol. 168 ofApplied Mathematical Sciences. Springer, 2010

    2010

  24. [24]

    Cantwell,Introduction to Symmetry Analysis

    B. Cantwell,Introduction to Symmetry Analysis. Cambridge Texts in Applied Mathematics. Cambridge University Press, 2002

    2002

  25. [25]

    Dimas and D

    S. Dimas and D. Tsoubelis,SYM: A new symmetry-finding package for Mathematica, inThe 10th international conference in modern group analysis, pp. 64–70, Nicosia, University of Cyprus, 2005

    2005

  26. [26]

    Olaussen, P

    K. Olaussen, P. Osland and M. A. Solberg,Symmetry and Mass Degeneration in Multi-Higgs-Doublet Models,JHEP07(2011) 020, [1007.1424]

    Pith/arXiv arXiv 2011

  27. [27]

    Coleman and E

    S. Coleman and E. Weinberg,Radiative corrections as the origin of spontaneous symmetry breaking,Phys. Rev. D7(1973) 1888–1910

    1973

  28. [28]

    Gildener and S

    E. Gildener and S. Weinberg,Symmetry breaking and scalar bosons,Phys. Rev. D 13(1976) 3333–3341

    1976

  29. [29]

    J. E. Camargo-Molina, B. O’Leary, W. Porod and F. Staub,Vevacious: A tool for finding the global minima of one-loop effective potentials,Eur. Phys. J. C73(2013) 2588, [1307.1477]

    Pith/arXiv arXiv 2013

  30. [30]

    Staub,Reopen parameter regions in two-higgs doublet models,Phys

    F. Staub,Reopen parameter regions in two-higgs doublet models,Phys. Lett. B776 (2018) 407–411, [1705.03677]

    Pith/arXiv arXiv 2018

  31. [31]

    M. E. Krauss, T. Opferkuch and F. Staub,The ultraviolet landscape of two-higgs doublet models,Eur. Phys. J. C78(2018) 1020, [1807.07581]

    Pith/arXiv arXiv 2018

  32. [32]

    I. P. Ivanov and F. Vaz˜ ao,Yet another lesson on the stability conditions in multi-higgs potentials,JHEP11(2020) 104, [2006.00036]

    Pith/arXiv arXiv 2020

  33. [33]

    Robens, T

    T. Robens, T. Stefaniak and J. Wittbrodt,Two-real-scalar-singlet extension of the sm: Lhc phenomenology and benchmark scenarios,The European Physical Journal C80(Feb., 2020)

    2020

  34. [34]

    Branco, P

    G. Branco, P. Ferreira, L. Lavoura, M. Rebelo, M. Sher and J. P. Silva,Theory and phenomenology of two-higgs-doublet models,Physics Reports516(July, 2012) 1–102

    2012

  35. [35]

    P. J. Olver,Equivalence, Invariants and Symmetry. Cambridge University Press, Cambridge, 1995, 10.1017/CBO9780511609565

    doi:10.1017/cbo9780511609565 1995

  36. [36]

    Andrada, M

    A. Andrada, M. L. Barberis, I. G. Dotti and G. P. Ovando,Product structures on four dimensional solvable lie algebras,Homology, Homotopy and Applications7 (2005) 9–37, [math/0402234]. 50

    Pith/arXiv arXiv 2005

  37. [37]

    Ovando,Four dimensional symplectic lie algebras,Beitr¨ age zur Algebra und Geometrie47(2006) 419–434

    G. Ovando,Four dimensional symplectic lie algebras,Beitr¨ age zur Algebra und Geometrie47(2006) 419–434

    2006

  38. [38]

    and Solberg, M

    Plantey, R. and Solberg, M. Aa.,Computable conditions for order-2 CP symmetry in NHDM potentials,JHEP05(2024) 260, [2404.02004]

    Pith/arXiv arXiv 2024

  39. [39]

    P. E. Hydon,How to use Lie symmetries to find discrete symmetries, inModern Group Analysis VII(N. H. Ibragimov, K. R. Naqvi and E. Straume, eds.), pp. 141–147. MARS Publishers, Trondheim, 1999

    1999

  40. [40]

    Hydon,How to construct the discrete symmetries of partial differential equations, European Journal of Applied Mathematics11(2000) 515–527

    P. Hydon,How to construct the discrete symmetries of partial differential equations, European Journal of Applied Mathematics11(2000) 515–527. 51

    2000