gl (3) polynomial integrable system: diﬀerent faces of the 3-body/ A 2 elliptic Calogero model

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Introduction
Let us take a finite-dimensional Lie algebra g spanned by the generators J i , i = 1, 2, . . ., dim g.The second degree polynomial in the J-generators, where {A, B} = AB + BA is the anti-commutator and {a}, {b} are parameters, defines the Hamiltonian of the quantum Euler-Arnold top in a constant magnetic field with components b i , i = 1, 2, . . ., dim g.It is well known that the generators J i of any semi-simple Lie algebra can be written in terms of the generators (p, q) of a Heisenberg algebra, hence, J i = J i (p, q).We call such a system a g-polynomial system if its Hamiltonian is defined as H(p, q) ≡ H(J(p, q)).
A particular example of an sl(2)-polynomial system was studied in details in [10, equation (13)], which is associated with the harmonic oscillator, H = −q p2 + q − p − 1 2 p = −J 0 J − + J 0 − p + 1 2 J − , where p = 0, 1 and are two sl(2) generators, J 0 , J − = −J − , see below.The general sl(2)-polynomial system is associated with the Heun operator, which is equivalent to the BC 1 elliptic Calogero model [11].The present paper is aimed at constructing an analogous but gl(3)-polynomial system starting from the quantum A 2 elliptic (3-body Calogero) model.Celebrated 3-body elliptic Calogero model or, stated differently, the A 2 elliptic model (in the Hamiltonian reduction nomenclature, see, e.g., [3]), describes three point-like one-dimensional particles of unit masses on the real line with pairwise interaction given by the Weierstrass ℘function with rectangular fundamental domain (ω, iω 1 ).It is characterized by the Hamiltonian H (e) where x 1 , x 2 , x 3 are the coordinates of the bodies, ∆ (3) is three-dimensional flat Laplace operator, which represents the kinetic energy, κ ≡ ν(ν − 1) is the coupling constant.The Weierstrass function ℘(x) ≡ ℘(x | g 2 , g 3 ) (see, e.g., [14]) is defined as the solution of the equation where g 2 , g 3 are the so-called elliptic invariants, which can be conveniently parameterized as follows: where τ , µ are parameters, and e 1 , e 2 , e 3 are its roots which are chosen, conventionally, to obey e ≡ e 1 + e 2 + e 3 = 0. Since the Hamiltonian (1.1) is translation-invariant, x → x + a, one can introduce the center-of-mass and relative coordinates, with the condition 3  1 y i = 0.The Laplacian ∆ (3) i in these coordinates takes the form, Separating out the center-of-mass coordinate Y , the two-dimensional Hamiltonian arises + ν(ν − 1)(℘(y 1 − y 2 ) + ℘(2y 1 + y 2 ) + ℘(y 1 + 2y 2 )), (1.4) which seemingly was already known to Charles Hermite as a two-dimensional generalization of the celebrated one-dimensional Lamé operator (following Serguei P. Novikov's studies of unpublished notes by Charles Hermite communicated to one of the authors (Alexander V. Turbiner)), H A 1 = − 1 2 which is also the Hamiltonian of the A 1 elliptic model [3], see also [8].We will call the operator (1.4) the two-dimensional Lamé operator.In general, the above procedure allows us to connect the quantum dynamics in the relative space of the three-body problem with twodimensional quantum dynamics [12].For many years the question of the existence of polynomial eigenfunctions of the operator (1.4) was a challenge to answer.It was eventually solved in 2015 by Sokolov-Turbiner in [6]: the discrete values of the coupling constant were found for which the (n+2)(n+1) 2 polynomial eigenfunctions exist in the variables where is the shifted Weierstrass function.
In very tedious and highly non-trivial calculations, performed in [6], it was found that the A 2 elliptic Calogero model potential V A 2 (see (1.1) and (1.4)) in variables (1.6) takes the form of ratio of polynomials, where the denominator was called the determinant or the elliptic discriminant.In rational limit τ = µ = 0 this is the square of Vandermonde determinant or the discriminant of the cubic equation, in trigonometric limit µ = 0 it becomes the so-called trigonometric Vandermonde determinant or a trigonometric discriminant, see for details [6].Furthermore, the two-dimensional flat Laplacian in (1.4) becomes the Laplace-Beltrami operator in (x, y)-coordinates with naturally-defined flat contravariant metric g ij , i, j = 1, 2 with polynomial entries.It can be easily checked that, remarkably, expression (1.7) is equal to the determinant of this contravariant metric, which explains the name determinant, used in [6].Surprisingly, the gauge rotation of the 2-dimensional Lamé operator (1.4) with the determinant D (1.7) to the power ν/2 as a gauge factor transforms operator (1.4) into the algebraic operator(!) with polynomial coefficients, This was one of the principal results obtained in the article [6], which will be essential in the present article.Let us emphasize that the operator h A 2 (x, y) looks like the two-dimensional generalization of the (algebraic) Heun operator, see, e.g., [11].It was also found in [6] that the second-order algebraic differential operator h A 2 (x, y) commutes with a non-trivial third-order algebraic differential operator k A 2 with polynomial coefficients, where Hence, h A 2 (x, y) and k A 2 (x, y) span the two-dimensional commutative algebra of the differential operators in two variables, which depend on three free parameters ν, µ, τ .This is the first nontrivial example of this.Naturally, the third-order differential operator k A 2 (x, y) can be called the integral.By making the inverse gauge rotation of the integral k A 2 (x, y), with the determinant D (1.7) to the power (−ν/2) as the gauge factor and changing variables (x, y) → (y 1 , y 2 ) (1.6), we should arrive at the third-order integral of the quantum 3-body elliptic Calogero model in the form of the third-order differential operator with elliptic coefficients found by Oshima [4].This demonstrates explicitly the integrability of the original 3-body elliptic Calogero model written in y 1 , y 2 coordinates.It was concluded in [6] that the 3-body elliptic Calogero model defines a polynomial integrable model with the algebraic Hamiltonian (1.8) and the algebraic integral (1.9) with µ, τ, νdependent parametric coefficients.This model has sl(3) hidden algebra in the representation (−3ν, 0).As a result the sl(3) quantum Euler-Arnold top in a constant magnetic field occurs.Note that for discrete values of the coupling constant κ: n = −3ν, n = 0, 1, 2, . . ., the sl(3) hidden algebra emerges in the finite-dimensional representation, thus, the top has a common finite-dimensional invariant subspace for both h and k.
The goal of this article is two-fold.First of all, the above-mentioned polynomial integrable model, realized in terms of differential operators, will be rewritten in terms of the generators of the Heisenberg algebra h 5 .Hence, its Hamiltonian will appear as an element of the universal enveloping algebra U h 5 .Then we project it into the translation-invariant or dilatation-invariant operators defining two families of 3-parametric µ, τ , ν isospectral polynomial integrable models on two-dimensional uniform or exponential lattices, respectively, and two additional families on mixed two-dimensional translation-invariant and dilatation-invariant lattices.All four families admit 2-parametric µ, τ polynomial eigenfunctions for certain discrete values of the coupling constant ν.An extra polynomial integrable model occurs as a result of a special complexification of R 2 to C 2 via the Heisenberg algebra h 5 generators acting on the two-dimensional Hilbert space with the Gaussian measure.The spectrum of this model is characterized by infinite multiplicity and for certain discrete values of the coupling constant κ (1.5) the eigenfunctions are polyanalytic functions in two complex variables of the fixed degree.Second of all, it will be shown that gl(3) polynomial integrable model, defined in the Fock space, is related with special bilinear and trilinear, 2-parametric elements of the universal enveloping algebra of the algebra gl(3).It turns out that these non-linear elements commute once they are written in terms of any concrete realization of the algebra gl(3) by elements of the universal enveloping algebra U h 5 .
The article is organized with introduction, Sections 2-7, conclusions and three Appendices A, B, C. In Section 2, the 3-body elliptic Calogero model in algebraic form is reformulated in Fock space and its gl(3)-polynomial integrable model is defined.Section 3 contains four lattice versions of the 3-body elliptic Calogero model.Section 4 is dedicated to complexification of the gl(3)-polynomial integrable model into C 2 .In Section 5, all nine artifacts of the gl(3) algebra are presented as bilinear combinations of the gl(3) generators and Theorem is proved that all of them vanish if the gl(3) generators are written as non-linear elements of the universal enveloping algebra U h 5 .Section 6 contains the explicit expressions of the Hamiltonian, the cubic integral and their commutator in terms of the gl(3)-algebra generators.In Section 7, the G 2 /3-body (with pairwise and 3-body interactions) elliptic problem is briefly discussed and the Fock space representation of the G 2 elliptic 3-body Hamiltonian is constructed.
Throughout the remaining text the hats in p, q's will be dropped: (p, q) → (p, q).

3-body elliptic Calogero model in the Fock space
Let us take 5-dimensional Heisenberg algebra h 5 spanned by the generators p x , p y , q x , q x , I, which obey the commutation relations, see Appendix A.3.The universal enveloping algebra of the algebra h 5 : U h 5 , is spanned by all ordered monomials in p x , p y , q x , q y .Now let us form in U h 5 a second degree polynomial in p-generators, h A 2 (p x , q x , p y , q y ) = q x + 3τ q 2 x + 3µq 3 x + 3 µ − τ 2 q 2 y − 3µτ q x q 2 y − 3µ 2 q 2 x q 2 y p 2 x where τ , µ, ν are parameters.Here the coefficients c ij are the 4th degree polynomials in qgenerators, c i are the 3rd degree ones and c 0 is the 2nd degree polynomial.We also form another non-linear combination in p, q-generators in the U h 5 , where the coefficients d ijk , d ij , d i , d 0 are polynomials in q of degrees 6, 5, 4 and 3, respectively.

Proof . By direct calculation. ■
Note, that it can be checked by direct calculation that h A 2 , k A 2 written in the (classical) phase space (p, q)-variables, {p, q} = 1, do not form the commutative pair with respect to the Poisson bracket, {h A 2 , k A 2 } ̸ = 0, for any values of the parameters τ , µ, ν.The reason for it is conceptual.Since we do not discuss in this paper the classical-quantum correspondence being fully focused on the quantum case, this will be discussed elsewhere.
By introducing the vacuum |0⟩ as an object annihilated by p-operators: in addition to the universal enveloping algebra U h 5 , this leads to definition of the Fock space.
The formal eigenvalue problem in the Fock space for the Hamiltonian h A 2 is as follows: where ϕ (h) (q) is the eigen-operator and λ (h) is the eigenvalue (spectral parameter).Analogously, Owing to Theorem 2.1 the eigenvalue problems (2.4) and (2.5) have common eigen-operators ϕ.
For n = 0 (thus, at zero coupling, κ = 0), For n = 1 at coupling the operator h A 2 has a three-dimensional kernel (three zero modes) of the type with coefficients a 1 , a 2 which do not vanish simultaneously and, The first non-zero eigenvalues appear for n = 2, thus, at κ = 10 9 .
In total, there exist six polynomial eigenstates.Eigenvalues are given by the roots of the factorized algebraic equation of degree 6, Explicitly, and the corresponding six eigen-operators are of the form a 1 q 2 x + a 2 q x q y + a 3 q 2 y b 1 q x + b 2 q y + c with parameters a 1 , a 2 , a 3 , which do not vanish simultaneously, and b 1 , b 2 , c.In the limit τ = µ = 0 (it corresponds to the rational A 2 integrable model without the harmonic oscillator terms) all six eigenvalues are degenerate to zero.

Uniform translation-invariant lattice
Let us introduce the shift operator, where δ ∈ R is parameter, which, sometimes, is called spacing, and construct a canonical pair of shift operators (see, e.g., [5]) where the operator D δ is defined as sometimes, it is called the Norlund derivative.It is easy to check that [D δ , X δ ] = 1, hence, D δ , X δ form the canonical pair, both operators are non-local.In the limit δ → 0 this pair becomes the well-known coordinate-momentum representation (∂ x , x) of the Heisenberg algebra h 3 (p, q, I), For non-vanishing δ, the canonical pair (3.1) belongs to the extended universal enveloping algebra Ûh 3 .These operators act on infinite uniform lattice space with spacing δ {. . ., x − 2δ, x − δ, x, x + δ, x + 2δ, . . .} marked by x ∈ R -a position of a central (or reference) point of the lattice.
By taking D δ , X δ (3.1) as basic elements, it can be shown that algebra h 5 of finite-difference (shift) operators can be formed: Evidently, the vacuum vector, This algebra acts on the rectangular uniform lattice with spacings (δ 1 , δ 2 ).By identifying in (2.2) and (2.3) the variables (p, q) with (D δ , X δ ), we arrive at the Hamiltonian and the integral of the polynomial integrable model on the two-dimensional uniform lattice with spacings (δ 1 , δ 2 ), and If parameter −ν = n/3, n = 0, 1, 2, . . ., the eigenvalue problems for the operators (3.3) and (3.4) have (n+2)(n+1) 2 common polynomial eigenfunctions ϕ (h,k) (x, y) in the form of triangular polynomials, The first polynomial eigenfunctions for n = 0, 1, 2 can be easily found by using the results collected in Examples 2.2.

Exponential dilatation-invariant lattice
Let us introduce the dilation operator, where q ∈ C, and construct a canonical pair of dilatation operators see [1], where [D q , X q ] = 1 for any q.It can be checked that their product is q-independent, The operator D q is called the Jackson symbol (or the Jackson derivative).Both operators X q , D q are pseudodifferential operators with action on monomials as follows: where {n} q = 1−q n 1−q is the so called q-number n.By taking D q , X q (3.5) as basic elements it can be shown that algebra h 5 of discrete operators can be formed: y ] = 0, [X q 1 ,x , X q 2 ,y ] = 0, [D q 1 (q 2 ),x(y) , I] = 0, [X q 1 (q 2 ),x(y) , I] = 0, cf.(3.2).Evidently, the vacuum vector, This algebra acts on the exponential lattice with spacings (q 1 , q 2 ).By identifying in (2.2) and (2.3) the variables (p, q) with (D q , X q ) we arrive at the Hamiltonian and the integral of the polynomial integrable model on the two-dimensional exponential lattice with spacings (q 1 , q 2 ), and common polynomial eigenfunctions ϕ (h,k) (x, y) in the form of triangular polynomials, The first polynomial eigenfunctions for n = 0, 1, 2 can be easily found by using the results collected in Examples 2.2.

Mixed translation-invariant and dilatation-invariant lattice
It is evident that one can construct the operators h, k acting in x-direction on the uniform lattice and in y-direction on the exponential lattice and visa versa.Therefore, there are two ways to realize it by taking or In both cases the vacuum vector remains the same, In a straightforward way one can build the Hamiltonian and the integral and for (3.8) and and

gl(3)-polynomial integrable model in C 2
Introduce the five-dimensional Heisenberg algebra h 5 ≡ H 5 = a 1 , a † 1 , a 2 , a † 2 , I with commutator a i , a † j = δ ij I, i, j = 1, 2, [a i , a j ] = a † i , a † j = 0 and [a i , I] = a † i , I = 0 by using a new, mathematics-oriented notations [13].Its representation on the standard Hilbert space, where dv(z) = dxdy is the Euclidean volume measure on C = R 2 , is given by two canonical pairs of raising and lowering operators related to z where a † j is adjoint to a j with a j , a † j = I, j = 1, 2, see [13] for details and I is the identity operator.The vacuum vector |0⟩, defined by is any two-dimensional analytic function.Formally, by taking (2.2) and (2.3) one can build the Hamiltonian and the integral It is evident that they continue to commute.This procedure can be considered as a complexification of the original polynomial model (1.8) and (1.9), which emerged from the 3-body/A 2 elliptic Calogero model as its algebraic version.Formally, the Hamiltonian is the sixth-order differential operator in ∂ ∂z , ∂ ∂ z .Note that the first polynomial eigenfunctions of h (C 2 ) for n = 0, 1, 2 can be easily found by using the results collected in Examples 2.2.

gl(3) algebra: artifacts
Long ago one of the authors (Alexander V. Turbiner) discovered in the algebra gl(3) with generators defined in (A.1) the existence of nine bilinear combinations in generators with unusual property: all those bilinear combinations vanish if the representation of gl(3) generators by the first-order differential operators (A.2) is taken!The explicit form of the bilinear combinations is the following [9]: (5.1) Theorem 5.1.For the gl(3) generators, written in terms of the Heisenberg algebra h 5 generators (A.3), all nine artifacts (5.1) vanish A 1,...,9 (p x,y , q x,y ) = 0.
This theorem can be proved by direct calculation.
It can be checked that the commutators of the artifacts are of the form where c k ij (J) for i ̸ = j are linear combinations of the gl(3) generators.For example, see Appendix C for the explicit calculation.Hence, A i do not span a Lie algebra.Interesting question to ask is what would happen with the artifacts A's if instead of gl(3) generators the gl(3) Kac-Moody currents are taken.It will be addressed elsewhere.
6 The Hamiltonian and the integral in gl(3)-algebra generators

Hamiltonian
By taking the Hamiltonian (2.2), one can demonstrate that it can be rewritten in the gl(3) abstract generators, which obey formally the commutation relations (A.1), hence, in extremely compact form; here µ, τ are parameters and the dependence on ν can be included into the representation (into the generators) and eventually is absent!Hence, (6.1) is two-parametric, bilinear element of the universal enveloping algebra U gl (3) .If µ = τ = 0 the element h A 2 (6.1) dramatically simplifies, By substituting the generators J 0,1,5,6 in the form of differential operators (A.2), one can see that this element corresponds to the 3-body/A 2 rational Calogero model (without harmonic oscillator term).Non-surprisingly, the raising generators J 7,8 are absent in this case, as well as the generators J 4,3,2 .

Cubic integral
In a similar way, as was done in order to construct (6.1), by taking the integral (2.3) in the Fock space representation one can demonstrate that it can be rewritten in the gl(3) abstract generators which obey the commutation relations (A.1), where µ, τ are parameters, see (1.2), and the explicit dependence on ν is absent!Hence, it is two-parametric, trilinear element of the universal enveloping algebra U gl (3) .If µ = τ = 0, the element k A 2 (6.2) dramatically simplifies, it corresponds to the 3-body/A 2 rational integrable Calogero model (without harmonic oscillator term).Since this is the exactly-solvable problem, non-surprisingly, the raising generators J 7,8 are absent.

Commutator
By taking (6.1) and ( 6.2), one can make the extremely cumbersome (and very lengthy) calculation of their Lie bracket (commutator) by using the specially designed Maple 18 code.An example of the code is presented in Appendix C. It was the main goal of the master thesis of one of the authors (Miguel A. Guadarrama-Ayala).Eventually, it leads to the following statement: Theorem 6.1.The commutator of the expressions (6.1) and (6.2) is the linear superposition of artifacts (5.1), for any values of parameters τ , µ, where c i (J) are some coefficient functions in J's.
Proof .By direct calculation by using the specially designed Maple 18 code.It was carried out on a regular DELL desktop computer with 2.4 GhZ working frequency and 6 GB RAM memory.Intuitively, this result (6.3) is evident: in the Fock space representation, where h, k ∈ U h 5 , the commutator should vanish, see Theorems 2.1 and 5.1.Hence, the commutator should be a (non)-linear combination of the artifacts.The fact that is a linear combination of the artifacts is non-trivial.
Alternative way to represent the commutator (6.3) is as follows where for the coefficients D(J, A) are presented in Appendix B. ■

G 2 elliptic 3-body problem
By adding the 3-body interaction potential to the 3-body elliptic Calogero Hamiltonian (1.4), we arrive at the 3-body Wolfes elliptic Hamiltonian in (y 1 , y 2 )-coordinates (1.3), which is also called the G 2 elliptic Hamiltonian in the Hamiltonian reduction nomenclature [3].
After extremely tedious (and very lengthy) symbolic calculations by using the Maple 18 code, see Appendix C for an example, it can be shown that the existence of a differential operator k m (u, v) of degree five such that the operator commutes with the G 2 elliptic Hamiltonian h G 2 (7.2); k m has the form of polynomial in λ of finite degree.Note that in the particular case of the G 2 rational Hamiltonian (see (7.2) at µ = τ = 0), this operator was calculated in [7] (where it corresponded to the case k = 3) in slightly different variables other than u, v: it is a polynomial in λ of degree four.In general, this operator will be presented in its explicit form elsewhere.So far this operator is unknown in the explicit form.By taking the 5-dimensional Heisenberg algebra h 5 spanned by the generators p u , p v , q u , q v , I, see (2.1), one can form the following second degree polynomial in p u , p v : + λ 6 1 + 2τ q u + µq 2 u p u + 4 −q 2 u + 3τ q v + 3µq u q v p v + 18νµq u .
It is easy to check that if (p, q)-variables are taken in the coordinate-momentum representation, cf. (A.4), the expression (7.3) is reduced to the operator (7.2).The operator h G 2 (p u , p v , q u , q v ) represents the G 2 elliptic model in the Fock space.By substituting into (7.3) the representations (A.5)-(A.6),we will arrive at the G 2 elliptic lattice Hamiltonians defined on uniform-uniform, uniform-exponential, exponential-uniform, exponential-exponential lattices in (u, v) space as well as the complexified G 2 elliptic Hamiltonian in the algebraic form.

Conclusions
In this paper, a polynomial integrable system, associated with the algebra U h 5 and inspired by the algebraic representation of the A 2 elliptic model in Fock space is defined.It has the form of a second degree polynomial in p i , i = 1, 2, for the Hamiltonian and a 3rd degree polynomial in p i , i = 1, 2, for the integral, where the coefficients {c} and {d} are polynomials in q of a finite degrees, while (p i , q i ) form a canonical pair.Overall, the operators h A 2 and k A 2 depend on three free parameters µ, τ , ν. Remarkably, both operators h A 2 and k A 2 can be rewritten in terms of the sl(3) generators J 1,2,...,8 and they can be embedded into the U h 5 algebra in the (−3ν, 0) representation (A.3).Hence, ν corresponds to the mark of the representation.
It can be conjectured that Conjecture 8.1.Up to canonical transformation p → p + f (q), q → q, there are no other non-trivial commuting operators in the U h 5 algebra of degree 2 and 3 in p other than h (8.1) and k (8.2).
The operators h and k can be rewritten in terms of the abstract gl(3) generators which obey the commutation relations (A.1) and which give a non-vanishing commutator [h, k].However, once the gl(3) generators are taken in the concrete representation (A.3) the operators h and k becomes h A 2 (2.2) and k A 2 (2.3), respectively, and their commutator [h A 2 , k A 2 ] = 0.The remarkable property of the commutator [h, k] is that it can be written as a linear superposition of the artifacts A 1,2,...,9 .We doubt there exist other elements of the universal enveloping algebra U gl(3) (up to automorphisms) with such a property.Different realizations of (p i , q i ), i = 1, 2 by differential operators, finite-difference operators, discrete operators, or the operators in z, z variables lead to a variety of concrete isospectral quantum integrable polynomial systems in two continuous variables, on two-dimensional uniform, exponential lattices or mixed ones, and on the C 2 complex space.All these integrable models depend on the continuous parameter ν.If this parameter takes certain discrete values, all above-mentioned integrable systems become quasi-exactly-solvable ones admitting a finite number of polynomial eigenfunctions in the form of triangular polynomials.

A.1 Structure constants
The commutation relations (A.1) of the gl(3) algebra can be represented as where c k ij are the structure constants.The non-vanishing structure constants are:

A.2 Representation of gl(3) algebra in differential operators
The algebra gl(3) with commutation relations (A.1) can be realized by the first-order differential operators in two variables, and where ν is parameter.It corresponds to the irreducible representation of the spin (−3ν, 0).If −3ν = n is integer, the finite-dimensional representation space which is spanned by triangular polynomials, where the coefficients D 1,...,12 are presented by superposition of the ordered polynomials in gl(3)-generators J 0,1,...,8 multiplied by the artifacts A 1,...,9 of the gl(3) algebra, %Commutator(j8, j7) = 0); Here we define the lexicographic ordering of the j-generators via the list "jorder" > jorder := [j8, j7, j6, j5, j4, j3, j2, j1, j0]: Here we define the function OR which orders the J-generators in every term of an expression in the lexicographic order defined above.
The function OR uses the command SortProducts which is a command of the Physics Library which acts on a Maple expression containing noncommutative products.It uses the list "jorder" which defines the ordering of the J-generators.The option "usecommutator" indicates that commutators defined previously.> OR(Simplify(CommutatorA1A2 -AnsatzcommutatorA1A2)); 0 Note added in proof.We know that A 2 rational system, see (1.8) for the Hamiltonian h 3 ≡ h A 2 (x, y) and (1.9) for the cubic integral k 3 ≡ k A 2 (x, y) in the algebraic form at µ = τ = 0, is integrable, [h 3 , k 3 ] = 0; however, this rational system admits separation of variables in polar coordinates in the space of relative motion and, hence, the existence of the additional quadratic integral of motion x 3 , see [7]: [h 3 , x 3 ] = 0 with property I ≡ [x 3 , k 3 ] ̸ = 0; hence, [h 3 , I] = 0. Thus, the A 2 rational integrable system is superintegrable.Recently, it was shown in [2] that double commutators [x 3 , I] = P 3 (h 3 , k 3 , x 3 , I) and [k 3 , I] = Q 3 (h 3 , k 3 , x 3 , I) are cubic polynomials in (h 3 , k 3 , x 3 , I).Hence, we arrive at 4-generated, infinite-dimensional cubic polynomial algebra of integrals.