Tremblay-Turbiner-Winternitz (TTW) system at integer index $k$: polynomial algebras of integrals

Alexander V Turbiner; Juan Carlos L\'opez Vieyra

arxiv: 2503.09502 · v4 · submitted 2025-03-12 · 🧮 math-ph · hep-th· math.MP· nlin.SI

Tremblay-Turbiner-Winternitz (TTW) system at integer index k: polynomial algebras of integrals

Juan Carlos L\'opez Vieyra , Alexander V Turbiner This is my paper

Pith reviewed 2026-05-23 00:26 UTC · model grok-4.3

classification 🧮 math-ph hep-thmath.MPnlin.SI

keywords TTW systemsuperintegrable systemspolynomial algebrashidden algebraexactly solvable modelsintegrals of motionquantum mechanics on plane

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The pith

For the TTW system at any integer k the Hamiltonian together with two integrals and their commutator generate a polynomial algebra of order k+1.

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

The paper studies the infinite family of superintegrable TTW quantum systems on the plane, parameterized by an integer index k, that separate in polar coordinates. Through direct computation for k=1,2,3,4 it verifies that the Hamiltonian H, the two integrals I1 and I2, and the commutator I12 satisfy commutation relations in which [I1,I12] and [I2,I12] equal polynomials of degree exactly k+1 in the four generators. These relations are presented as evidence for an underlying hidden algebra g^(k) whose finite-dimensional representations contain the system's invariant subspaces. The authors conjecture that the same four-element polynomial algebra structure and the hidden algebra exist for every positive integer k, so that the full algebra of integrals is realized as a subalgebra of g^(k).

Core claim

It is conjectured that for any integer k the Hamiltonian H, two integrals I1,2 and their commutator I12 are four generating elements of the polynomial algebra of integrals of the order (k+1): [I1,I12]=P_{k+1}(H,I1,2,I12), [I2,I12]=Q_{k+1}(H,I1,2,I12) with P and Q polynomials of degree k+1. This structure is linked to a hidden algebra g^(k) that possesses finite-dimensional representation spaces as invariant subspaces, and explicit checks for k=1,2,3,4 confirm both the polynomial relations and the presence of g^(k). The conjecture asserts that the polynomial algebra of integrals is a subalgebra of g^(k) and that the pattern continues without modification for all positive integers k.

What carries the argument

The four generators H, I1, I2, I12 of the polynomial algebra of order k+1, together with the hidden algebra g^(k) whose finite-dimensional representations furnish the invariant subspaces.

If this is right

The commutators of the integrals close inside the span of ordered monomials of total degree at most k+1 in the four generators.
The complete algebra of all integrals of the TTW system is realized as a subalgebra inside the hidden algebra g^(k).
Finite-dimensional representations of g^(k) supply the invariant subspaces needed for exact solvability at every integer k.
The same four-element generating set works uniformly for the entire infinite family indexed by k.

Where Pith is reading between the lines

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

A uniform algebraic construction might now be used to produce the integrals for arbitrary k without case-by-case computation.
The same hidden-algebra mechanism could be tested on other families of superintegrable systems that separate in polar coordinates.
If the conjecture holds, the dimension of the space of integrals of a given degree would be controlled by the representation theory of g^(k).

Load-bearing premise

The pattern of polynomial relations and the existence of the hidden algebra g^(k) observed by explicit computation for k=1,2,3,4 continues to hold without modification or exceptions for every positive integer k.

What would settle it

An explicit calculation for k=5 in which the commutator [I1,I12] cannot be expressed as any polynomial of degree 6 in the ordered monomials of H, I1, I2, I12 would disprove the conjecture.

read the original abstract

An infinite 3-parametric family of superintegrable and exactly-solvable quantum models on a plane, admitting separation of variables in polar coordinates, marked by integer index $k$ was introduced in Journ Phys A 42 (2009) 242001 and was called in literature the TTW system. In this paper it is conjectured that the Hamiltonian and both integrals of TTW system have hidden algebra $g^{(k)}$ - it was checked for $k=1,2,3,4$ - having its finite-dimensional representation spaces as the invariant subspaces. It is checked that for $k=1,2,3,4$ that the Hamiltonian $H$, two integrals ${\cal I}_{1,2}$ and their commutator ${\cal I}_{12} = [{\cal I}_1,{\cal I}_2]$ are four generating elements of the polynomial algebra of integrals of the order $(k+1)$: $[{\cal I}_1,{\cal I}_{12}] = P_{k+1}(H, {\cal I}_{1,2},{\cal I}_{12})$, $[{\cal I}_2,{\cal I}_{12}] = Q_{k+1}(H, {\cal I}_{1,2},{\cal I}_{12})$, where $P_{k+1},Q_{k+1}$ are polynomials of degree $(k+1)$ written in terms of ordered monomials of $H, {\cal I}_{1,2},{\cal I}_{12}$. This implies that polynomial algebra of integrals is subalgebra of $g^{(k)}$. It is conjectured that all is true for any integer $k$.

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 conjectures that the TTW integrals generate a polynomial algebra of order k+1 inside a hidden algebra g^(k) for any integer k, but only verifies the pattern for k=1-4.

read the letter

The core claim is a conjecture: for integer k the Hamiltonian H and integrals I1, I2, I12 generate a polynomial algebra of order k+1, and this algebra sits inside the hidden algebra g^(k). The authors verify the commutator relations explicitly for k=1,2,3,4 and note that the 2009 TTW construction already supplies the base system. The explicit commutator calculations for those four cases are the concrete new content; they show the polynomials P_{k+1} and Q_{k+1} in ordered monomials of the generators. Those calculations appear reproducible from the given expressions and do not rely on fitting or circular definitions. The citation to the original TTW work is appropriate and the framing stays within the established program of algebraic structures for superintegrable systems. The obvious limitation is the jump from four checked values to the general statement for arbitrary k. No inductive step, representation-theoretic argument, or general derivation is supplied, so the result remains a pattern observed in small cases rather than a proven theorem. The hidden algebra g^(k) itself is introduced via the checks and not derived independently. This work is aimed at the small group of people already working on the algebraic side of 2D superintegrable systems who might want to prove or disprove the conjecture. It does not resolve broader questions about classification or quantization. I would not bring it to a general reading group, would not cite it in my own papers without a proof, and would send it to peer review only in a specialized journal where such conjectures with explicit checks are routinely considered.

Referee Report

1 major / 2 minor

Summary. The manuscript conjectures that for the TTW superintegrable system with integer index k the Hamiltonian H, integrals I1 and I2, and their commutator I12 are four generating elements of a polynomial algebra of integrals of order (k+1), satisfying [I1,I12]=P_{k+1}(H,I1,I2,I12) and [I2,I12]=Q_{k+1}(H,I1,I2,I12) with P_{k+1},Q_{k+1} polynomials of degree k+1 in ordered monomials. It further conjectures that these elements generate a hidden algebra g^{(k)} whose finite-dimensional representations are invariant subspaces and that the polynomial algebra is a subalgebra of g^{(k)}. Both statements are verified by direct symbolic computation for k=1,2,3,4 and asserted to hold for every positive integer k.

Significance. If the conjecture holds it would supply a uniform algebraic description of the integrals for the entire TTW family and relate them to a hidden algebra g^{(k)}, which could aid classification of superintegrable systems and spectral analysis. The explicit commutator calculations for k=1–4 constitute concrete, machine-verifiable evidence for the polynomial relations in those cases and are a clear strength of the work.

major comments (1)

Abstract (final sentence) and concluding paragraph: the claim that the structure holds for arbitrary integer k is supported solely by explicit verification for k=1,2,3,4; this limited base is load-bearing for the general statement and leaves open the possibility that the pattern of relations or the hidden algebra g^{(k)} fails to persist for larger k. A concrete test (e.g., the commutator calculation for k=5) or an inductive argument would be required to elevate the conjecture beyond the checked cases.

minor comments (2)

The explicit forms of the polynomials P_{k+1} and Q_{k+1} are referenced but not displayed; including at least the lowest-order cases (k=1) in the main text would improve clarity.
Notation for the ordered monomials in the polynomial algebra is used without a dedicated definition paragraph; a short notational summary would reduce ambiguity.

Simulated Author's Rebuttal

1 responses · 1 unresolved

We thank the referee for their thorough review and for recognizing the significance of the explicit computations for k=1 to 4. We address the major comment below.

read point-by-point responses

Referee: [—] Abstract (final sentence) and concluding paragraph: the claim that the structure holds for arbitrary integer k is supported solely by explicit verification for k=1,2,3,4; this limited base is load-bearing for the general statement and leaves open the possibility that the pattern of relations or the hidden algebra g^{(k)} fails to persist for larger k. A concrete test (e.g., the commutator calculation for k=5) or an inductive argument would be required to elevate the conjecture beyond the checked cases.

Authors: The manuscript presents the general statement explicitly as a conjecture, verified computationally for k=1,2,3,4. We do not assert a proof for arbitrary k but observe that the polynomial relations and the hidden algebra structure hold consistently in the checked cases, suggesting the pattern persists. We agree that a general proof or additional verification would be valuable. However, extending the symbolic computation to k=5 involves significantly higher computational complexity due to the increasing degree of the polynomials, which was beyond the scope of the present work. We will revise the abstract and concluding paragraph to more explicitly highlight that the general case is conjectural and based on the observed pattern for small k. revision: partial

standing simulated objections not resolved

Providing an inductive argument or explicit verification for k=5 to prove the conjecture for arbitrary k.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper frames its central result explicitly as a conjecture, supported by direct symbolic commutator calculations for the finite cases k=1,2,3,4. No load-bearing step reduces by definition, by renaming a fitted input as a prediction, or by a self-citation chain whose cited result itself depends on the present claim. The 2009 reference is used only to introduce the Hamiltonian family; the polynomial-algebra structure and hidden algebra g^(k) are verified independently in the checked cases and then conjectured to persist. This is the normal, non-circular pattern for an explicit-computation-plus-conjecture manuscript.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the unproven generalization from four explicit cases to all integer k; no free parameters are fitted, but the hidden algebra g^(k) is introduced without independent evidence beyond the small-k checks.

axioms (1)

domain assumption The operators H, I1, I2 satisfy the stated commutation relations that close into a polynomial algebra of degree k+1.
Invoked when the authors assert that the four elements generate the algebra for general k.

invented entities (1)

hidden algebra g^(k) no independent evidence
purpose: To serve as the ambient algebra containing the polynomial subalgebra generated by the integrals.
Postulated to explain the structure observed in the small-k calculations; no independent falsifiable prediction is given.

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discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Quantum two-dimensional superintegrable systems in flat space: exact-solvability, hidden algebra, polynomial algebra of integrals
math-ph 2025-12 unverdicted novelty 2.0

Review confirms exact solvability, algebraic forms, hidden Lie algebras, and polynomial integral algebras for six 2D superintegrable systems including Smorodinsky-Winternitz, Fokas-Lagerstrom, Calogero-Wolfes, and TTW models.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages · cited by 1 Pith paper · 1 internal anchor

[1]

An infinite family of solvable and integrable quantum systems on a plane

F. Tremblay, A.V. Turbiner and P. Winternitz, An infinite family of solvable and integrable quantum systems on a plane, Journal of Phys.A42(2009) 242001 (10 pp), math-ph arXiv:0904.0738v4 (extended)

work page internal anchor Pith review Pith/arXiv arXiv 2009
[2]

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J. Wolfes, On the three-body linear problem with three-body interaction, J. Math. Phys.15, 1420-1424 (1974)

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F. Calogero, Solution of a three-body problem in one dimension, J. Math. Phys.10, 2191–2196 (1969) 22

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M.A. Olshanetsky and A. M. Perelomov, Quantum completely integrable systems connected with semi-simple Lie algebras, Lett. Math. Phys.2(1977) 7–13

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Olshanetsky and A.M

M.A. Olshanetsky and A.M. Perelomov, Quantum integrable systems related to Lie algebras, Phys. Repts.94(1983) 313-393

work page 1983
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L´ opez Vieyra and A.V

J.C. L´ opez Vieyra and A.V. Turbiner, Wolfes model akaG 2/I6-rational integrable model:g (2), g(3) hidden algebras and quartic poly- nomial algebra of integrals, J. Math. Phys.65, 051702 (2024)

work page 2024
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Turbiner, colloquium talk to the memory of Willard Miller Jr, given at Universidad Complutense de Madrid, June 6, 2024

A.V. Turbiner, colloquium talk to the memory of Willard Miller Jr, given at Universidad Complutense de Madrid, June 6, 2024

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Kalnins, W

E. Kalnins, W. Miller Jr. and S. Post, Coupling constant metamorphosis and Nth order symmetries in classical and quantum me- chanics, J. Phys. A: Math. Theor.43035202 (2010)

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Kalnins, J

E. Kalnins, J. M. Kress, W. Miller Jr., A Recurrence Relation Approach to Higher Order Quantum Superintegrability, SIGMA7, 031 (2011) (24 pp.)

work page 2011
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Rosenbaum, A.V

M. Rosenbaum, A.V. Turbiner and A. Capella, Solvability of theG 2 integrable system, Intern.Journ.Mod.Phys.A13(1998), 3885-3904

work page 1998
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Boreskov, A.V

K.G. Boreskov, A.V. Turbiner and J.C. L´ opez Vieyra Solvability of the Hamiltonians Related to Exceptional Root Spaces: Rational Case, Comm Math. Phys260, 17 (2005)

work page 2005
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Turbiner, From quantumA N (Sutherland) toE 8 trigonometric model: space-of-orbits view, SIGMA9, 003 (2013)

A.V. Turbiner, From quantumA N (Sutherland) toE 8 trigonometric model: space-of-orbits view, SIGMA9, 003 (2013)

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Lie, Theorie der Transformationsgruppen, Math

S. Lie, Theorie der Transformationsgruppen, Math. Ann.16(1880) 441-528;Gesammelte Abhandlungen, vol.6 23 (B. G. Teubner, Leipzig, 1927), pp. 1-94; R. Hermann and M. Ackerman,Sophus Lie’s 1880 transformation group paper (Math. Sci. Press, Brookline, Mass., 1975) (english translation)

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Gonz´ alez-L´ opez, N

A. Gonz´ alez-L´ opez, N. Kamran and P.J. Olver, Lie Algebras of Vector Fields in the Real Plane, Proc. Lond. Math. Soc.64(1992) 339–368

work page 1992
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Gonz´ alez-L´ opez, N

A. Gonz´ alez-L´ opez, N. Kamran and P.J. Olver, New quasi-exactly-solvable Hamiltonians in two dimensions, Comm.Math.Phys.,159(1994) 503-537

work page 1994
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Turbiner,Hidden Algebra of Three-Body Integrable Systems, Mod.Phys.Lett.A13, 1473-1483 (1998)

A. Turbiner,Hidden Algebra of Three-Body Integrable Systems, Mod.Phys.Lett.A13, 1473-1483 (1998)

work page 1998
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Turbiner, TalkA new family of planar solvable and integrable Schroedinger equations, given at Instituto de Ciencias Nucleares, UNAM, August 18, 2010

A.V. Turbiner, TalkA new family of planar solvable and integrable Schroedinger equations, given at Instituto de Ciencias Nucleares, UNAM, August 18, 2010

work page 2010
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Smirnov and A.V

Yu.F. Smirnov and A.V. Turbiner, gln+1 algebra of matrix differential operators and matrix quasi-exactly-solvable problems, Acta Polytechnica53, 462-469 (2013)

work page 2013
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Fri˘s, V

J. Fri˘s, V. Mandrosov, Ya.A. Smorodinsky, M. Uhlir and P. Winternitz, On higher symmetries in quantum mechanics, Phys.Lett.16, 354-356 (1965)

work page 1965
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Winternitz, Ya.A

P. Winternitz, Ya.A. Smorodinsky, M. Uhlir and J. Fri˘s, Symmetry groups in classical and quantum mechanics, Yad. Fiz.4, 625-635 (1966); Sov.Journ.Nucl.Phys4, 444-450 (1967) (English Translation)

work page 1966
[21]

Evans, Group theory of the Smorodinsky-Winternitz system, J

N.W. Evans, Group theory of the Smorodinsky-Winternitz system, J. Math. Phys.32, 3369-3375 (1991)

work page 1991
[22]

Tempesta, A

P. Tempesta, A. Turbiner and P. Winternitz, Exact Solvability of Superintegrable Systems, J. Math. Phys.424248-4257 (2001) 24

work page 2001
[23]

Post, Models of Quadratic Algebras Generated by Superintegrable Systems in 2D, SIGMA7, 036 (2011) (20 pp.)

S. Post, Models of Quadratic Algebras Generated by Superintegrable Systems in 2D, SIGMA7, 036 (2011) (20 pp.)

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Tremblay, A.V

F. Tremblay, A.V. Turbiner and P. Winternitz, Periodic orbits for an infinite family of classical superintegrable systems, Journal of PhysicsA43015202 (2010) 25 Appendix A: TTW atk= 3: the second integralI 2 coefficients, see (36) Non-vanishing coefficients for the integralI 2 of TTW atk= 3, defined in (36): C (2) 0,1 =−1296(2a+ 1)(2b+ 1)(a+b+ 2)(3a+ 3b+ ...

work page 2010

[1] [1]

An infinite family of solvable and integrable quantum systems on a plane

F. Tremblay, A.V. Turbiner and P. Winternitz, An infinite family of solvable and integrable quantum systems on a plane, Journal of Phys.A42(2009) 242001 (10 pp), math-ph arXiv:0904.0738v4 (extended)

work page internal anchor Pith review Pith/arXiv arXiv 2009

[2] [2]

Wolfes, On the three-body linear problem with three-body interaction, J

J. Wolfes, On the three-body linear problem with three-body interaction, J. Math. Phys.15, 1420-1424 (1974)

work page 1974

[3] [3]

Calogero, Solution of a three-body problem in one dimension, J

F. Calogero, Solution of a three-body problem in one dimension, J. Math. Phys.10, 2191–2196 (1969) 22

work page 1969

[4] [4]

Olshanetsky and A

M.A. Olshanetsky and A. M. Perelomov, Quantum completely integrable systems connected with semi-simple Lie algebras, Lett. Math. Phys.2(1977) 7–13

work page 1977

[5] [5]

Olshanetsky and A.M

M.A. Olshanetsky and A.M. Perelomov, Quantum integrable systems related to Lie algebras, Phys. Repts.94(1983) 313-393

work page 1983

[6] [6]

L´ opez Vieyra and A.V

J.C. L´ opez Vieyra and A.V. Turbiner, Wolfes model akaG 2/I6-rational integrable model:g (2), g(3) hidden algebras and quartic poly- nomial algebra of integrals, J. Math. Phys.65, 051702 (2024)

work page 2024

[7] [7]

Turbiner, colloquium talk to the memory of Willard Miller Jr, given at Universidad Complutense de Madrid, June 6, 2024

A.V. Turbiner, colloquium talk to the memory of Willard Miller Jr, given at Universidad Complutense de Madrid, June 6, 2024

work page 2024

[8] [8]

Kalnins, W

E. Kalnins, W. Miller Jr. and S. Post, Coupling constant metamorphosis and Nth order symmetries in classical and quantum me- chanics, J. Phys. A: Math. Theor.43035202 (2010)

work page 2010

[9] [9]

Kalnins, J

E. Kalnins, J. M. Kress, W. Miller Jr., A Recurrence Relation Approach to Higher Order Quantum Superintegrability, SIGMA7, 031 (2011) (24 pp.)

work page 2011

[10] [10]

Rosenbaum, A.V

M. Rosenbaum, A.V. Turbiner and A. Capella, Solvability of theG 2 integrable system, Intern.Journ.Mod.Phys.A13(1998), 3885-3904

work page 1998

[11] [11]

Boreskov, A.V

K.G. Boreskov, A.V. Turbiner and J.C. L´ opez Vieyra Solvability of the Hamiltonians Related to Exceptional Root Spaces: Rational Case, Comm Math. Phys260, 17 (2005)

work page 2005

[12] [12]

Turbiner, From quantumA N (Sutherland) toE 8 trigonometric model: space-of-orbits view, SIGMA9, 003 (2013)

A.V. Turbiner, From quantumA N (Sutherland) toE 8 trigonometric model: space-of-orbits view, SIGMA9, 003 (2013)

work page 2013

[13] [13]

Lie, Theorie der Transformationsgruppen, Math

S. Lie, Theorie der Transformationsgruppen, Math. Ann.16(1880) 441-528;Gesammelte Abhandlungen, vol.6 23 (B. G. Teubner, Leipzig, 1927), pp. 1-94; R. Hermann and M. Ackerman,Sophus Lie’s 1880 transformation group paper (Math. Sci. Press, Brookline, Mass., 1975) (english translation)

work page 1927

[14] [14]

Gonz´ alez-L´ opez, N

A. Gonz´ alez-L´ opez, N. Kamran and P.J. Olver, Lie Algebras of Vector Fields in the Real Plane, Proc. Lond. Math. Soc.64(1992) 339–368

work page 1992

[15] [15]

Gonz´ alez-L´ opez, N

A. Gonz´ alez-L´ opez, N. Kamran and P.J. Olver, New quasi-exactly-solvable Hamiltonians in two dimensions, Comm.Math.Phys.,159(1994) 503-537

work page 1994

[16] [16]

Turbiner,Hidden Algebra of Three-Body Integrable Systems, Mod.Phys.Lett.A13, 1473-1483 (1998)

A. Turbiner,Hidden Algebra of Three-Body Integrable Systems, Mod.Phys.Lett.A13, 1473-1483 (1998)

work page 1998

[17] [17]

Turbiner, TalkA new family of planar solvable and integrable Schroedinger equations, given at Instituto de Ciencias Nucleares, UNAM, August 18, 2010

A.V. Turbiner, TalkA new family of planar solvable and integrable Schroedinger equations, given at Instituto de Ciencias Nucleares, UNAM, August 18, 2010

work page 2010

[18] [18]

Smirnov and A.V

Yu.F. Smirnov and A.V. Turbiner, gln+1 algebra of matrix differential operators and matrix quasi-exactly-solvable problems, Acta Polytechnica53, 462-469 (2013)

work page 2013

[19] [19]

Fri˘s, V

J. Fri˘s, V. Mandrosov, Ya.A. Smorodinsky, M. Uhlir and P. Winternitz, On higher symmetries in quantum mechanics, Phys.Lett.16, 354-356 (1965)

work page 1965

[20] [20]

Winternitz, Ya.A

P. Winternitz, Ya.A. Smorodinsky, M. Uhlir and J. Fri˘s, Symmetry groups in classical and quantum mechanics, Yad. Fiz.4, 625-635 (1966); Sov.Journ.Nucl.Phys4, 444-450 (1967) (English Translation)

work page 1966

[21] [21]

Evans, Group theory of the Smorodinsky-Winternitz system, J

N.W. Evans, Group theory of the Smorodinsky-Winternitz system, J. Math. Phys.32, 3369-3375 (1991)

work page 1991

[22] [22]

Tempesta, A

P. Tempesta, A. Turbiner and P. Winternitz, Exact Solvability of Superintegrable Systems, J. Math. Phys.424248-4257 (2001) 24

work page 2001

[23] [23]

Post, Models of Quadratic Algebras Generated by Superintegrable Systems in 2D, SIGMA7, 036 (2011) (20 pp.)

S. Post, Models of Quadratic Algebras Generated by Superintegrable Systems in 2D, SIGMA7, 036 (2011) (20 pp.)

work page 2011

[24] [24]

Tremblay, A.V

F. Tremblay, A.V. Turbiner and P. Winternitz, Periodic orbits for an infinite family of classical superintegrable systems, Journal of PhysicsA43015202 (2010) 25 Appendix A: TTW atk= 3: the second integralI 2 coefficients, see (36) Non-vanishing coefficients for the integralI 2 of TTW atk= 3, defined in (36): C (2) 0,1 =−1296(2a+ 1)(2b+ 1)(a+b+ 2)(3a+ 3b+ ...

work page 2010