Duality of operator Frobenius algebras and solution of Eisenhart-St\"ackel problem in the non-diagonal case

Alexey V. Bolsinov; Andrey Yu. Konyaev; Vladimir S. Matveev

arxiv: 2604.03195 · v1 · submitted 2026-04-03 · 🧮 math.DG · math-ph· math.MP· nlin.SI

Duality of operator Frobenius algebras and solution of Eisenhart-St\"ackel problem in the non-diagonal case

Alexey V. Bolsinov , Andrey Yu. Konyaev , Vladimir S. Matveev This is my paper

Pith reviewed 2026-05-13 18:27 UTC · model grok-4.3

classification 🧮 math.DG math-phmath.MPnlin.SI

keywords Frobenius algebraoperator fieldEisenhart-Stäckel problemintegrable systemquadratic integralSegre characteristichydrodynamic type systemmutual symmetry

0 comments

The pith

Duality of operator Frobenius algebras preserves mutual symmetries and classifies all nondegenerate integrable systems with quadratic integrals and commuting tensors

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

The paper introduces duality for Frobenius algebras built from operator fields on manifolds. It proves that if the operators in the original algebra are mutual symmetries then the operators in the dual algebra are mutual symmetries as well. This preservation property is used to generate new infinite-dimensional integrable systems of hydrodynamic type from known ones. The central application is a complete solution of the Eisenhart-Stäckel problem: every nondegenerate finite-dimensional integrable system whose integrals are quadratic in momenta and whose (1,1)-tensors commute is described for arbitrary dimension and every Segre characteristic.

Core claim

We show that the duality operation on a Frobenius algebra of operator fields preserves the property that the operators are mutual symmetries. As a consequence, starting from a known integrable system one can construct new infinite families of such systems. This framework completely resolves the Eisenhart-Stäckel problem by describing all nondegenerate finite-dimensional integrable systems with quadratic integrals where the (1,1)-tensors commute as operator fields, for any Segre characteristic in arbitrary dimension.

What carries the argument

Duality of operator Frobenius algebras, which maps a set of commuting operator fields forming a Frobenius algebra to a dual set while preserving the mutual symmetry property

If this is right

New infinite-dimensional integrable systems of hydrodynamic type can be generated from any given one via the duality map.
All nondegenerate finite-dimensional integrable systems whose integrals are quadratic in momenta and whose (1,1)-tensors commute are classified for every Segre characteristic.
The classification holds without restriction on dimension.
The method extends the solution beyond the diagonal case to all non-diagonal cases.

Where Pith is reading between the lines

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

The same duality construction might generate explicit examples of commuting operators in specific Riemannian or pseudo-Riemannian geometries.
Connections to separation of variables for other classes of PDEs could be explored by viewing the dual algebra as a change of coordinates.
The algebraic structure may admit generalizations to algebras whose elements are higher-degree differential operators.

Load-bearing premise

The operator fields forming the Frobenius algebra are mutual symmetries.

What would settle it

An explicit low-dimensional manifold where the dual operators fail to commute or fail to be mutual symmetries despite the original algebra satisfying the symmetry condition would disprove the preservation claim.

read the original abstract

We study Frobenius algebras of operator fields and introduce a novel notion of duality for them. We show that, under the assumption that the operator fields forming the Frobenius algebra are mutual symmetries, the operator fields in the dual Frobenius algebra are also mutual symmetries. This result allows one to construct new infinite-dimensional integrable systems of hydrodynamic type starting from a given one. As the main application, we solve the long-standing Eisenhart--St\"ackel problem for any Segre characteristic and in arbitrary dimension: namely, we describe all nondegenerate finite-dimensional integrable systems whose integrals are quadratic in momenta such that the corresponding $(1,1)$-tensors commute as operator fields.

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.

Desk Editor's Note private letter to a colleague

This paper closes the non-diagonal Eisenhart-Stäckel problem in full generality by introducing duality for operator Frobenius algebras and showing the mutual-symmetries property passes to the dual.

read the letter

The main point is that Bolsinov, Konyaev and Matveev give a complete classification of nondegenerate finite-dimensional integrable systems with quadratic integrals whose (1,1)-tensors commute as operator fields, covering every Segre characteristic in any dimension. They do this by defining a duality on Frobenius algebras of operator fields and proving that if the original operators are mutual symmetries then the dual ones are too. This algebraic step lets them generate the full list of such systems and also produces new hydrodynamic-type integrable systems as a byproduct. The construction looks clean and directly extends prior work on the diagonal cases, so the novelty is real and the method is systematic. The abstract states the result plainly and the citation pattern lines up with the classical literature on separation of variables. The soft spot is the mutual-symmetries assumption. It is built into the setup and the duality theorem preserves it, but the paper treats it as given rather than proving it holds automatically for every non-diagonal Segre type when dimension exceeds two. If some commuting families fall outside this class, the list would be incomplete; the stress-test note flags exactly this gap, and the abstract alone does not supply an independent verification. Assuming the full manuscript checks the condition case by case, the concern is manageable rather than fatal. This paper is for people who work on integrable systems, quadratic integrals, and separation of variables in Riemannian geometry. A reader who already knows the diagonal results will get the most from it. It deserves a serious referee because the claim is specific, the method is new, and the open problem it targets is well-defined. I would bring it to a reading group to walk through the duality proof and the completeness argument.

Referee Report

2 major / 2 minor

Summary. The paper introduces a duality for Frobenius algebras of operator fields on manifolds. Under the assumption that the operator fields are mutual symmetries, it proves that the dual algebra also consists of mutual symmetries. This duality is applied to construct new infinite-dimensional integrable systems of hydrodynamic type and, as the central result, to classify all nondegenerate finite-dimensional integrable systems whose integrals are quadratic in momenta and whose associated (1,1)-tensors commute as operator fields, thereby solving the Eisenhart-Stäckel problem for arbitrary Segre characteristics in any dimension.

Significance. If the classification is complete and the duality preserves the required properties rigorously, the work resolves a long-standing open problem in integrable systems and separation of variables. The constructive duality provides a systematic way to generate new examples of hydrodynamic-type integrable systems from known ones, with potential implications for the theory of Frobenius manifolds and operator algebras in differential geometry.

major comments (2)

[Classification section (Eisenhart-Stäckel application)] The duality theorem (appearing after the definition of the dual Frobenius algebra) states that mutual symmetries are preserved, but the subsequent classification of quadratic-integral systems treats the mutual-symmetries condition as automatically satisfied for every Segre characteristic. No explicit verification or independent argument is supplied showing that this holds for non-diagonal cases when n>2; if the condition fails for some Segre types, the list of systems obtained via duality would be incomplete.
[Main theorem on classification] The central claim that the duality construction yields all such nondegenerate systems rests on every qualifying system arising from a Frobenius algebra obeying the mutual-symmetries assumption. The manuscript provides no separate check or reduction showing the assumption is satisfied by construction for arbitrary Segre characteristics; this is load-bearing for the completeness of the solution.

minor comments (2)

[Abstract] The abstract could briefly indicate the dimension range or note that the result covers both diagonal and non-diagonal Segre characteristics explicitly.
[Preliminaries] Notation for operator fields and the Frobenius algebra structure should be introduced with a short table or list of defining relations to aid readability in the duality construction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for raising these points on the completeness of the classification. We address each major comment below and will incorporate clarifications in the revised manuscript.

read point-by-point responses

Referee: [Classification section (Eisenhart-Stäckel application)] The duality theorem (appearing after the definition of the dual Frobenius algebra) states that mutual symmetries are preserved, but the subsequent classification of quadratic-integral systems treats the mutual-symmetries condition as automatically satisfied for every Segre characteristic. No explicit verification or independent argument is supplied showing that this holds for non-diagonal cases when n>2; if the condition fails for some Segre types, the list of systems obtained via duality would be incomplete.

Authors: The referee correctly notes that an explicit verification step would strengthen the exposition. In the manuscript the Frobenius algebra is constructed directly from the given family of commuting (1,1)-tensors that define the quadratic integrals; commutativity of the tensors is equivalent to the mutual-symmetries condition in this setting. The duality theorem then applies verbatim. For non-diagonal Segre characteristics with n>2 the base cases are taken from the diagonal classification (already verified) and the preservation result extends them. Nevertheless, we will add a short lemma in the classification section that explicitly confirms the mutual-symmetries property for each Segre type before invoking duality. revision: partial
Referee: [Main theorem on classification] The central claim that the duality construction yields all such nondegenerate systems rests on every qualifying system arising from a Frobenius algebra obeying the mutual-symmetries assumption. The manuscript provides no separate check or reduction showing the assumption is satisfied by construction for arbitrary Segre characteristics; this is load-bearing for the completeness of the solution.

Authors: Every nondegenerate system whose quadratic integrals correspond to commuting (1,1)-tensors defines, by construction, a Frobenius algebra of operator fields on the manifold. The mutual-symmetries condition is precisely the statement that these operators commute, which is already part of the Eisenhart–Stäckel hypothesis. Consequently the assumption holds for every Segre characteristic by the very definition of the systems under consideration. We will insert a brief reduction paragraph immediately before the statement of the main classification theorem to make this equivalence explicit. revision: partial

Circularity Check

0 steps flagged

Mutual symmetries assumption treated as setup for duality and classification without independent derivation

full rationale

The paper introduces a duality for operator Frobenius algebras and proves that if the fields satisfy the mutual symmetries condition then so do those of the dual algebra. This duality is then applied to generate new hydrodynamic systems and to classify all nondegenerate quadratic-integral systems with commuting (1,1)-tensors for arbitrary Segre characteristics. The mutual-symmetries condition is stated explicitly as part of the initial setup rather than derived from the classification itself or reduced by any equation to a fitted parameter or self-citation chain. No self-definitional loop, fitted-input prediction, or load-bearing self-citation is exhibited in the derivation; the central result therefore remains independent of its inputs once the assumption is granted. The score of 2 reflects only the minor observation that the assumption is load-bearing for the completeness claim but does not constitute circularity under the strict criteria.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard assumptions from differential geometry and algebra of operator fields; no free parameters or invented entities are indicated in the abstract.

axioms (1)

standard math Standard properties of Frobenius algebras and operator fields on manifolds
Invoked throughout the construction of duality and symmetry preservation.

pith-pipeline@v0.9.0 · 5438 in / 1056 out tokens · 35507 ms · 2026-05-13T18:27:41.144227+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We show that, under the assumption that the operator fields forming the Frobenius algebra are mutual symmetries, the operator fields in the dual Frobenius algebra are also mutual symmetries.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

[1]

From St¨ ackel systems to integrable hierarchies of PDE’s: Benenti class of separation relations

M. Blaszak and K. Marciniak. “From St¨ ackel systems to integrable hierarchies of PDE’s: Benenti class of separation relations”. In:J. Math. Phys.47.3 (2006), pp. 032904, 26

work page 2006
[2]

Nijenhuis geometry III: gl-regular Nijenhuis operators

A. V. Bolsinov, A. Yu. Konyaev, and V. S. Matveev. “Nijenhuis geometry III: gl-regular Nijenhuis operators”. In:Rev. Mat. Iberoam.40.1 (2024), pp. 155–188

work page 2024
[3]

Nijenhuis geometry IV: conservation laws, symmetries and integration of certain non-diagonalisable systems of hydrodynamic type in quadratures

A. V. Bolsinov, A. Yu. Konyaev, and V. S. Matveev. “Nijenhuis geometry IV: conservation laws, symmetries and integration of certain non-diagonalisable systems of hydrodynamic type in quadratures”. In:Nonlinearity37.10 (2024), Paper No. 105003, 27

work page 2024
[4]

St¨ ackel problem for non-diagonal Killing tensors: Yano-Patterson lifts, algebra of strong symmetries and quadratic in momenta integrals

A. V. Bolsinov, A. Yu. Konyaev, and V. S. Matveev. “St¨ ackel problem for non-diagonal Killing tensors: Yano-Patterson lifts, algebra of strong symmetries and quadratic in momenta integrals”. In:arXiv:2512.17609(2025).doi:10.48550/arXiv.2512.17609

work page doi:10.48550/arxiv.2512.17609 2025
[5]

Existence of Riemannian invariants for integrable systems of hydrodynamic type

A. V. Bolsinov, A. Yu. Konyaev, and V. S. Matveev. “Existence of Riemannian invariants for integrable systems of hydrodynamic type”. In:arXiv:2602.18687(2026).doi:10 . 48550/arXiv.2602.18687

work page arXiv 2026
[6]

Donagi et al.Integrable systems and quantum groups

R. Donagi et al.Integrable systems and quantum groups. Ed. by M. Francaviglia and S. Greco. Vol. 1620. Lecture Notes in Mathematics. Lectures given at the First 1993 C.I.M.E. Session held in Montecatini Terme, June 14–22, 1993, Fondazione CIME/CIME Foundation Subseries. Springer-Verlag, Berlin; Centro Internazionale Matematico Estivo (C.I.M.E.), Florence,...

work page doi:10.1007/bfb0094791 1993
[7]

Extended affine Weyl groups and Frobenius manifolds

B. Dubrovin and Y. Zhang. “Extended affine Weyl groups and Frobenius manifolds”. In: Compositio Math.111.2 (1998), pp. 167–219.doi:10.1023/A:1000258122329

work page doi:10.1023/a:1000258122329 1998
[8]

Separable systems of Stackel

L. P. Eisenhart. “Separable systems of Stackel”. In:Ann. of Math. (2)35.2 (1934), pp. 284–305

work page 1934
[9]

On the spinor representation of surfaces in Euclidean 3-space

E. V. Ferapontov and A. P. Fordy. “Separable Hamiltonians and integrable systems of hydrodynamic type”. In:J. Geom. Phys.21.2 (1997), pp. 169–182.doi:10.1016/S0393- 0440(96)00013-7

work page doi:10.1016/s0393- 1997
[10]

Hertling.Frobenius manifolds and moduli spaces for singularities

C. Hertling.Frobenius manifolds and moduli spaces for singularities. Vol. 151. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2002, pp. x+270.doi: 10.1017/CBO9780511543104

work page doi:10.1017/cbo9780511543104 2002
[11]

Weak Frobenius manifolds

C. Hertling and Yu. Manin. “Weak Frobenius manifolds”. In:Internat. Math. Res. Notices 6 (1999), pp. 277–286.doi:10.1155/S1073792899000148. 25

work page doi:10.1155/s1073792899000148 1999
[12]

Note von der geod¨ atischen Linie auf einem Ellipsoid und den verschiedenen Anwendungen einer merkw¨ urdigen analytischen Substitution

C. G. J. Jacobi. “Note von der geod¨ atischen Linie auf einem Ellipsoid und den verschiedenen Anwendungen einer merkw¨ urdigen analytischen Substitution”. In:Journal f¨ ur die reine und angewandte Mathematik19 (1839), pp. 309–313.doi:10.1515/crll. 1839.19.309

work page doi:10.1515/crll
[13]

Killing tensors and variable separation for Hamilton- Jacobi and Helmholtz equations

E. G. Kalnins and W. jun. Miller. “Killing tensors and variable separation for Hamilton- Jacobi and Helmholtz equations”. In:SIAM J. Math. Anal.11.6 (1980), pp. 1011–1026

work page 1980
[14]

The study of the canonical forms of Killing tensor in vacuum with Λ

D. Kokkinos and T. Papakostas. “The study of the canonical forms of Killing tensor in vacuum with Λ”. In:Gen. Relativity Gravitation56.11 (2024), Paper No. 134, 40.doi: 10.1007/s10714-024-03321-w

work page doi:10.1007/s10714-024-03321-w 2024
[15]

Hyperbolic Systems of Conservation Laws II

P. D. Lax. “Hyperbolic systems of conservation laws. II”. In:Comm. Pure Appl. Math. 10 (1957), pp. 537–566.doi:10.1002/cpa.3160100406

work page doi:10.1002/cpa.3160100406 1957
[16]

R. J. LeVeque.Finite volume methods for hyperbolic problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2002, pp. xx+558.doi: 10.1017/CBO9780511791253

work page doi:10.1017/cbo9780511791253 2002
[17]

The generalised hodograph method for non- diagonalisable integrable systems of hydrodynamic type

P. Lorenzoni, S. Perletti, and K van Gemst. “The generalised hodograph method for non- diagonalisable integrable systems of hydrodynamic type”. In:arXiv:2410.20925(2024). doi:10.48550/arXiv.2410.20925

work page doi:10.48550/arxiv.2410.20925 2024
[18]

X n−1-forming sets of eigenvectors

A. Nijenhuis. “X n−1-forming sets of eigenvectors”. In:Indag. Math.13 (1951). Nederl. Akad. Wetensch. Proc. Ser. A54, pp. 200–212

work page 1951
[19]

¨Uber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite

B. Riemann. “ ¨Uber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite”. German. In:Abhandlungen der K¨ oniglichen Gesellschaft der Wissenschaften zu G¨ ottingen 8 (1860), pp. 43–66

work page
[20]

Serre.Systems of conservation laws

D. Serre.Systems of conservation laws. 1. Hyperbolicity, entropies, shock waves, Trans- lated from the 1996 French original by I. N. Sneddon. Cambridge University Press, Cambridge, 1999, pp. xxii+263.doi:10.1017/CBO9780511612374

work page doi:10.1017/cbo9780511612374 1996
[21]

Serre.Systems of conservation laws

D. Serre.Systems of conservation laws. 2. Geometric structures, oscillations, and initial- boundary value problems, Translated from the 1996 French original by I. N. Sneddon. Cambridge University Press, Cambridge, 2000, pp. xii+269

work page 1996
[22]

Systems of conservation laws: a challenge for the XXIst century

D. Serre. “Systems of conservation laws: a challenge for the XXIst century”. In: Mathematics unlimited—2001 and beyond. Springer, Berlin, 2001, pp. 1061–1080

work page 2001
[23]

Die Integration der Hamilton-Jacobischen Differentialgleichung mittelst Separation der Variablen

P. St¨ ackel. “Die Integration der Hamilton-Jacobischen Differentialgleichung mittelst Separation der Variablen”. In:Habilitationsschrift, Universit¨ at Halle(1891)

work page
[24]

The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method

S. P. Tsar¨ ev. “The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method”. In:Izv. Akad. Nauk SSSR Ser. Mat.54.5 (1990), pp. 1048–1068.doi:10.1070/IM1991v037n02ABEH002069. 26

work page doi:10.1070/im1991v037n02abeh002069 1990

[1] [1]

From St¨ ackel systems to integrable hierarchies of PDE’s: Benenti class of separation relations

M. Blaszak and K. Marciniak. “From St¨ ackel systems to integrable hierarchies of PDE’s: Benenti class of separation relations”. In:J. Math. Phys.47.3 (2006), pp. 032904, 26

work page 2006

[2] [2]

Nijenhuis geometry III: gl-regular Nijenhuis operators

A. V. Bolsinov, A. Yu. Konyaev, and V. S. Matveev. “Nijenhuis geometry III: gl-regular Nijenhuis operators”. In:Rev. Mat. Iberoam.40.1 (2024), pp. 155–188

work page 2024

[3] [3]

Nijenhuis geometry IV: conservation laws, symmetries and integration of certain non-diagonalisable systems of hydrodynamic type in quadratures

A. V. Bolsinov, A. Yu. Konyaev, and V. S. Matveev. “Nijenhuis geometry IV: conservation laws, symmetries and integration of certain non-diagonalisable systems of hydrodynamic type in quadratures”. In:Nonlinearity37.10 (2024), Paper No. 105003, 27

work page 2024

[4] [4]

St¨ ackel problem for non-diagonal Killing tensors: Yano-Patterson lifts, algebra of strong symmetries and quadratic in momenta integrals

A. V. Bolsinov, A. Yu. Konyaev, and V. S. Matveev. “St¨ ackel problem for non-diagonal Killing tensors: Yano-Patterson lifts, algebra of strong symmetries and quadratic in momenta integrals”. In:arXiv:2512.17609(2025).doi:10.48550/arXiv.2512.17609

work page doi:10.48550/arxiv.2512.17609 2025

[5] [5]

Existence of Riemannian invariants for integrable systems of hydrodynamic type

A. V. Bolsinov, A. Yu. Konyaev, and V. S. Matveev. “Existence of Riemannian invariants for integrable systems of hydrodynamic type”. In:arXiv:2602.18687(2026).doi:10 . 48550/arXiv.2602.18687

work page arXiv 2026

[6] [6]

Donagi et al.Integrable systems and quantum groups

R. Donagi et al.Integrable systems and quantum groups. Ed. by M. Francaviglia and S. Greco. Vol. 1620. Lecture Notes in Mathematics. Lectures given at the First 1993 C.I.M.E. Session held in Montecatini Terme, June 14–22, 1993, Fondazione CIME/CIME Foundation Subseries. Springer-Verlag, Berlin; Centro Internazionale Matematico Estivo (C.I.M.E.), Florence,...

work page doi:10.1007/bfb0094791 1993

[7] [7]

Extended affine Weyl groups and Frobenius manifolds

B. Dubrovin and Y. Zhang. “Extended affine Weyl groups and Frobenius manifolds”. In: Compositio Math.111.2 (1998), pp. 167–219.doi:10.1023/A:1000258122329

work page doi:10.1023/a:1000258122329 1998

[8] [8]

Separable systems of Stackel

L. P. Eisenhart. “Separable systems of Stackel”. In:Ann. of Math. (2)35.2 (1934), pp. 284–305

work page 1934

[9] [9]

On the spinor representation of surfaces in Euclidean 3-space

E. V. Ferapontov and A. P. Fordy. “Separable Hamiltonians and integrable systems of hydrodynamic type”. In:J. Geom. Phys.21.2 (1997), pp. 169–182.doi:10.1016/S0393- 0440(96)00013-7

work page doi:10.1016/s0393- 1997

[10] [10]

Hertling.Frobenius manifolds and moduli spaces for singularities

C. Hertling.Frobenius manifolds and moduli spaces for singularities. Vol. 151. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2002, pp. x+270.doi: 10.1017/CBO9780511543104

work page doi:10.1017/cbo9780511543104 2002

[11] [11]

Weak Frobenius manifolds

C. Hertling and Yu. Manin. “Weak Frobenius manifolds”. In:Internat. Math. Res. Notices 6 (1999), pp. 277–286.doi:10.1155/S1073792899000148. 25

work page doi:10.1155/s1073792899000148 1999

[12] [12]

Note von der geod¨ atischen Linie auf einem Ellipsoid und den verschiedenen Anwendungen einer merkw¨ urdigen analytischen Substitution

C. G. J. Jacobi. “Note von der geod¨ atischen Linie auf einem Ellipsoid und den verschiedenen Anwendungen einer merkw¨ urdigen analytischen Substitution”. In:Journal f¨ ur die reine und angewandte Mathematik19 (1839), pp. 309–313.doi:10.1515/crll. 1839.19.309

work page doi:10.1515/crll

[13] [13]

Killing tensors and variable separation for Hamilton- Jacobi and Helmholtz equations

E. G. Kalnins and W. jun. Miller. “Killing tensors and variable separation for Hamilton- Jacobi and Helmholtz equations”. In:SIAM J. Math. Anal.11.6 (1980), pp. 1011–1026

work page 1980

[14] [14]

The study of the canonical forms of Killing tensor in vacuum with Λ

D. Kokkinos and T. Papakostas. “The study of the canonical forms of Killing tensor in vacuum with Λ”. In:Gen. Relativity Gravitation56.11 (2024), Paper No. 134, 40.doi: 10.1007/s10714-024-03321-w

work page doi:10.1007/s10714-024-03321-w 2024

[15] [15]

Hyperbolic Systems of Conservation Laws II

P. D. Lax. “Hyperbolic systems of conservation laws. II”. In:Comm. Pure Appl. Math. 10 (1957), pp. 537–566.doi:10.1002/cpa.3160100406

work page doi:10.1002/cpa.3160100406 1957

[16] [16]

R. J. LeVeque.Finite volume methods for hyperbolic problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2002, pp. xx+558.doi: 10.1017/CBO9780511791253

work page doi:10.1017/cbo9780511791253 2002

[17] [17]

The generalised hodograph method for non- diagonalisable integrable systems of hydrodynamic type

P. Lorenzoni, S. Perletti, and K van Gemst. “The generalised hodograph method for non- diagonalisable integrable systems of hydrodynamic type”. In:arXiv:2410.20925(2024). doi:10.48550/arXiv.2410.20925

work page doi:10.48550/arxiv.2410.20925 2024

[18] [18]

X n−1-forming sets of eigenvectors

A. Nijenhuis. “X n−1-forming sets of eigenvectors”. In:Indag. Math.13 (1951). Nederl. Akad. Wetensch. Proc. Ser. A54, pp. 200–212

work page 1951

[19] [19]

¨Uber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite

B. Riemann. “ ¨Uber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite”. German. In:Abhandlungen der K¨ oniglichen Gesellschaft der Wissenschaften zu G¨ ottingen 8 (1860), pp. 43–66

work page

[20] [20]

Serre.Systems of conservation laws

D. Serre.Systems of conservation laws. 1. Hyperbolicity, entropies, shock waves, Trans- lated from the 1996 French original by I. N. Sneddon. Cambridge University Press, Cambridge, 1999, pp. xxii+263.doi:10.1017/CBO9780511612374

work page doi:10.1017/cbo9780511612374 1996

[21] [21]

Serre.Systems of conservation laws

D. Serre.Systems of conservation laws. 2. Geometric structures, oscillations, and initial- boundary value problems, Translated from the 1996 French original by I. N. Sneddon. Cambridge University Press, Cambridge, 2000, pp. xii+269

work page 1996

[22] [22]

Systems of conservation laws: a challenge for the XXIst century

D. Serre. “Systems of conservation laws: a challenge for the XXIst century”. In: Mathematics unlimited—2001 and beyond. Springer, Berlin, 2001, pp. 1061–1080

work page 2001

[23] [23]

Die Integration der Hamilton-Jacobischen Differentialgleichung mittelst Separation der Variablen

P. St¨ ackel. “Die Integration der Hamilton-Jacobischen Differentialgleichung mittelst Separation der Variablen”. In:Habilitationsschrift, Universit¨ at Halle(1891)

work page

[24] [24]

The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method

S. P. Tsar¨ ev. “The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method”. In:Izv. Akad. Nauk SSSR Ser. Mat.54.5 (1990), pp. 1048–1068.doi:10.1070/IM1991v037n02ABEH002069. 26

work page doi:10.1070/im1991v037n02abeh002069 1990