Superintegrability of Calogero-Moser systems associated with the cyclic quiver

We study complex integrable systems on quiver varieties associated with the cyclic quiver, and prove their superintegrability by explicitly constructing first integrals. We interpret them as rational Calogero-Moser systems endowed with internal degrees of freedom called spins. They encompass the usual systems in type $A_{n-1}$ and $B_n$, as well as generalisations introduced by Chalykh and Silantyev in connection with the multicomponent KP hierarchy. We also prove that superintegrability is preserved when a harmonic oscillator potential is added.


Introduction
The integrable n-particle systems of Toda [38], Calogero-Moser [7,30], and Ruijsenaars-Schneider [35] have a remarkable tendency to maintain many of their interesting properties when being extended in various ways 1 . These properties include superintegrability (if present) and their connections with one another as well as with other objects, may those be soliton equations, orthogonal polynomials or models in statistical physics. The extensions we have in mind include giving the particles internal degrees of freedom (spin models), replacing the underlying type A root system (boundary potentials) or defining the systems on exotic spaces (e.g. quiver varieties). This paper reinforces the above-mentioned phenomenon by proving the superintegrability of (spin) Calogero-Moser type systems attached to cyclic quivers.
Before delving into the particulars of the systems we are to study, let us define what we mean by superintegrability. For our purposes, a superintegrable Hamiltonian system with N degrees of freedom, that is a 2N -dimensional symplectic manifold (M, ω) with a smooth function H ∈ C ∞ (M ) of special importance, has 2N − 1 globally defined, independent constants of motion. Such systems are usually referred to as maximally superintegrable in the literature [43]. We note that maximal superintegrability is a special form of non-commutative (or degenerate) integrability [31,29]. The study of superintegrable systems has a long history with such notable examples as the Kepler problem or the n-dimensional isotropic harmonic oscillator [33], but despite its maturity, the field continues to furnish new developments, see e.g. [5,15,18,22,39].
The motivation for this work comes from Chalykh and Silantyev's paper [12] which generalised the KP hierarchy and (spin) Calogero-Moser type systems to cyclic quivers. A natural question to ask is: Are these new quiver generalisations of (spin) Calogero-Moser systems superintegrable? Our main result is an affirmative answer to this question via an explicit construction.
To help place this work into context, let us give a quick (incomplete) review of previous results on the superintegrability of (spin) CM systems. In 1975/76 Adler [1] showed the superintegrability of the rational Calogero-Moser Hamiltonian with a harmonic potential added (this variant is also known as the Calogero model). In 1983 Wojciechowski [44] proved superintegrability of all Hamiltonians of the rational Calogero-Moser system. In 1988 Ruijsenaars [36] published his scattering theory of rational and hyperbolic CM and RS systems (which implies superintegrability). In 1999 Caseiro-Françoise-Sasaki [9] proved superintegrability of rational CM attached to any finite Coxeter group. In 2003 Reshetikhin [34] established the degenerate integrability of spin CM systems corresponding to co-adjoint orbits of simple Lie algebras [28]. Let us also mention the papers [2,19] where explicitly formulated constants of motion for the rational RS system were found.
To give a sense of the type of integrable systems we consider, they include (as a special case) the rational B n spin Calogero-Moser model with an external harmonic oscillator potential whose Hamiltonian reads with particle momenta and positions (p i , x i ), spin variables f ij and arbitrary coupling constants γ 1 , ω. 1 For a brief overview of these integrable systems, we refer to the introductions of our PhD Theses [17,23]. 1 Note that the variables f ij can be seen as "collective" spins. For a fixed d > 1, they depend on 2nd (constrained) parameters that are interpreted as n sets of 2d spin variables, where one such set is attached to each particle.
The key idea (inspired by the works [2,3,8]) that lets us construct the constants of motion required for superintegrability can be summarised as follows. Let M be an arbitrary Poisson manifold (either real or complex) with a Poisson bracket {−, −}. Then we have the following Theorem 1.1. Fix a function H on M , and assume that there exists a family of functions (g j ) j∈N such that for all j ∈ N for some constants α j . a) For any j, k ∈ N with α j = α k , the function is a first integral of H. b) For any j ∈ N, the functionC is a first integral of H.
for some constants α j ,α j . Then, for any j, k ∈ N with α j = −α k , the function is a first integral of H.
The proofs of these results involve a straightforward use of the Leibniz rule and the assumptions. In fact, Theorems 1.1 and 1.2 hold more generally for derivations, so they can be used in the quantum case, too. Remark 1.3. In this paper, we adopt the convention N = {0, 1, . . .} and work in the complex setting, that is over the field of complex numbers C.
The structure of the paper is as follows. In Section 2, we describe the spinless Calogero-Moser spaces and prove superintegrability for spinless rational Calogero-Moser systems attached to cyclic quivers. Section 3 contains the spin generalisation of the results of Section 2. In Section 4, we prove superintegrability for the (spin) rational Calogero Hamiltonian (i.e. CM particles in a harmonic well) associated with classical Lie algebras. Section 5 explains the basics of the main computational tool of the paper, double brackets, and it contains the detailed derivations of formulas used in previous sections. Finally, in Section 6, we conclude the paper with an outlook on possible generalisations and future plans.
Acknowledgements. We thank L. Fehér for bringing relevant references to our attention. The work of M.F. was partly supported by a Rankin-Sneddon Research Fellowship of the University of Glasgow. This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No 795471.

Calogero-Moser system for the cyclic quiver
In this section, we consider Calogero-Moser spaces of complex dimension 2n associated with cyclic quivers on m ≥ 1 vertices extended by one arrow. Their connection to integrable systems in the simplest case (m = 1) goes back to Wilson [41], and has been extended in [12,24].
2.1. Description of the space. We omit a detailed introduction to the spaces at hand since they are special cases of the spaces introduced in Section 3. In those notations, we consider d = (1, 0, . . . , 0), and put V := V 0,1 and W := W 0,1 .
We fix integers n, m ≥ 1, and for I = Z/mZ we choose a generic λ = (λ s ) ∈ C I , see § 3.1 for the precise genericity conditions. We let | λ| = s∈I λ s . The Calogero-Moser space C n is obtained by Hamiltonian reduction from the set of matrices X s , Y s ∈ Mat n×n (C), s ∈ I = Z/mZ, V ∈ Mat 1×n (C), W ∈ Mat n×1 (C) , by requiring the n matrix conditions before considering orbits of the action of GL(n) = s∈I GL n (C) given by We consider a first restriction to the subset C • n ⊂ C n where the product X 0 . . . X m−1 ∈ Mat n×n (C) is diagonalisable, and its diagonal form is given by diag( We then choose a representative where X s = D for each s ∈ I, with D = diag(x 1 , . . . , x n ). Finally, we look at the subset C ′ n ⊂ C • n where for such representatives, the vector W has non-zero entries. In C ′ n , it is an easy exercise to see that we can parametrise any point (X s , Y s , V, W ) using where (x 1 , . . . , x n ) ∈ C n reg and (p 1 , . . . , p n ) ∈ C n . We can also see that this is unique up to Z m ≀ S n action, which acts by permutation of the entries using S n , and by (x 1 , . . . , x n ) → (µ r x 1 , . . . , µ r x n ) using Z m , where µ is a primitive m-th root of unity. The reduced Poisson bracket is canonical and given by 2.2. Superintegrability. We form the matrix X ∈ Mat nm×nm (C) as an m × m matrix with blocks of size n × n, where the only nonzero blocks are given by placing X s in position (s, s + 1). In the same way, we form Y ∈ Mat nm×nm (C) with only nonzero blocks being Y s placed in position (s + 1, s). (With the notations of § 3.1, X = s X s and Y = s Y s .) In particular, X k and Y k are block diagonal if and only if k is divisible by m. The functions tr Y mi , i ∈ N, are trivially Poisson commuting on C n , see Lemma 5.3. In this section, we are interested in proving that each such function is superintegrable based on the following example.
Example 2.1. In the case m = 1, we have for h i = 1 i tr Y i , that the functions (h i ) n i=1 define an integrable system such that h 2 is the Hamiltonian for the CM system. We note that for any i ∈ N × is a first integral of the integrable system. Thus is also a first integral of h i by Theorem 1.1. This is Wojciechowski's integral K (i) j+1,k+1 [44]. We now fix m ≥ 1, and set h m,i = 1 mi tr Y mi . Lemma 2.2. Fix i ∈ N × . For any j, k ∈ N, the function is a first integral of h m,i .
Proof. It is proved in Lemma 5.3 that {h m,i , tr(XY jm+1 )} = − tr Y m(i+j) , which is a first integral. So the result follows from Theorem 1.1 a). Proof. It suffices to show that h m,1 , . . . , h m,n and C m,i 2,1 , . . . , C m,i n,1 are functionally independent. This can be done as in [44], see the beginning of the proof of Proposition 3.6.
Remark 2.4. The fact that these systems are Liouville integrable appears in [12,Section V], and it is mentioned in [25, §4.4] for λ = (0, . . . , 0). Superintegrability in the case m = 1 corresponds to the original work of Wojciechowski [44]. Indeed, the function is the usual rational CM Hamiltonian of type A n−1 . The case m = 2 is equivalent to the B n case [12, which is the rational CM Hamiltonian in type B n , or type D n if λ 1 = 0 [32]. Superintegrability of rational CM systems associated with arbitrary root systems is established in [9].
3. Spin Calogero-Moser systems for the cyclic quiver 3.1. Phase space. We now define the general Calogero-Moser spaces associated with cyclic quivers. When there are several framing arrows going either to one vertex of the cyclic quiver, or when the number of framing arrows is the same for all the vertices in the cyclic quiver, these spaces and the corresponding integrable systems were first studied 2 in [12] and [24]. In the case m = 1, the spaces can be traced back to the works [42,6,37], where it was established that the systems correspond to the spin CM system due to Gibbons and Hermsen [26]. Fix an integer m ≥ 1 and let I = Z m = Z/mZ. When we consider I as a set, we identify it with {0, . . . , m − 1} by sending an element s ∈ I to its representative in {0, We consider the cyclic quiver on m arrows with framing corresponding to d, which is defined in the following way. Let Q d be the quiver with vertex set I = I ∪ {∞}, and whose edge set consists, for all s ∈ I, of d s + 1 arrows given by x s : s → s + 1 and v s,α : ∞ → s with α = 1, . . . , d s . (There is no arrow ∞ → s when d s = 0.) The doubleQ d of Q d then consists of the same vertex set I, and 2m + 2|d| arrows given by the ones described above together with y s = x * s : s + 1 → s, w s,α = v * s,α : s → ∞ for all 1 ≤ α ≤ d s and s ∈ I.
Remark 3.1. We adopt the following conventions for the rest of the text. The indices r, s range over I. When we consider a couple (s, α), for example as index of v s,α , we assume that s ∈ I as we have just explained and α ranges over the set {1, . . . , d s }. We omit such couples when d s = 0.
3.1.1. Definition of the space. We fix n = (n, 1) with n = (n s ) ∈ N I such that |n| = s n s > 0. A point ρ ∈ Rep(CQ d , n) consists of the vector space V = (⊕ s∈I V s ) ⊕ V ∞ with V s = C ns for each s ∈ I and V ∞ = C, together with 2m + 2|d| matrices given by which respectively represent the arrows x s , y s , v s,α , w s,α . We identify the point ρ with the tuple of matrices (X s , Y s , V s,α , W s,α ) to ease our discussion. We directly see that Rep(CQ d , n) is a smooth affine variety of dimension 2 s∈I n s (n s+1 + d s ).
We have a GL(n) := s∈I GL ns (C) action on Rep(CQ d , n) given by Following e.g. Van den Bergh [40], the complex manifold Rep(CQ d , n) admits a Poisson bracket {−, −} given by and which is zero on any other pair of entries of the matrices (3.1). Moreover, it is endowed with a moment map µ with value in gl(n) := s∈I gl ns (C) given by where we omit the final sum in µ s if d s = 0. Fix λ = (λ s ) ∈ C I and denote by λ · Id ∈ gl(n) the element with blocks λ s Id ns ∈ gl ns (C). Then, the slice µ −1 ( λ · Id) corresponds to imposing the m equations from which it follows by taking traces that s∈I 1≤α≤ds V s,α W s,α = − s λ s n s =: − λ · n. Using Hamiltonian reduction, it follows that the GIT quotient C n,d, λ = µ −1 ( λ · Id)// GL(n) is a Poisson variety. The space hence obtained is a quiver variety : it is the GIT quotient for the GL(n) action (3.2) on the representation space associated with a deformed preprojective algebra of Q with parameter ( λ, − λ · n).
From now on, we further assume that n = (n, . . . , n) for some n ∈ N × , and we simply denote C n,d, λ by C n . Then, C n is a non-empty smooth variety which coincides with the set-theoretic orbit space µ −1 ( λ · Id)/ GL(n) provided that the regularity conditions are satisfied, see [4,Proposition 3] or [14,Theorem 1.2]. Note that C n has dimension 2n|d|.

Local description. We consider the open subspace
We pick m-th roots (x i ) of the eigenvalues, and by construction of C • n these elements take value in C n reg (2.3). We can use the GL(n) action to pick any representative such that X s = diag(x 1 , . . . , x n ) for each s ∈ I, and there remains an overall action by the normaliser N of the diagonal subgroup (C × ) n ⊂ GL n (C) seen as a subgroup of GL(n) through N ∋ h → s∈I h ∈ GL(n).
We then define the open subspace C ′ n ⊂ C • n where for one (hence any) such representative, the vector 1≤α≤d0 W 0,α has non-zero entries. We can then act by a diagonal matrix to find a representative such that 1≤α≤d0 W 0,α = (1, . . . , 1) ⊤ . This representative is unique up to a Z m ≀ S n action described below. Note that C ′ n contains the subspace defined in § 2.1. In this way, we can characterise a point of C ′ n by the 2n + 2n|d| variables (x i , p i , v s,α,i , w s,α,i ) such that (x i ) ∈ C n reg , together with the 2n constraints by considering the following matrices This choice is unique up to Z m ≀ S n action, which acts by permutation of the entries using S n , and by (x 1 , . . . , x n ) → (µ r x 1 , . . . , µ r x n ) using Z m where µ is a primitive m-th root of unity. It is easy to see that we have the normalisation Proof. This result is a direct application of Lemma 5.4. To see this, we note that the following expressions can be written in terms of the local variables on C ′ n tr X km =m n j=1 In particular, we have that It is then a standard computation to see that (5.14a)-(5.14b) written in coordinates yield (3.10a). After these identities are established, we also get from (5.14c)-(5.14d) that (3.10b) holds. Next, using (5.14e) with r = r ′ = 0 and summing over all α, α ′ ∈ {1, . . . , d 0 }, we find the identity (3.10c). Taking r ′ = 0 and summing over α ′ also in (5.14e), we find the first identity in (3.10d). Finally, we can use these Poisson brackets and (5.14e) for arbitrary r, r ′ to obtain the second equality in (3.10d).    Proof. Let us use the full notation C n,d, λ of the space to emphasise the dependence on the framing d and the parameter λ. We form d • = (d m−1 , . . . , d 0 ) and λ • = − λ, noting that λ • satisfies (3.6) just asλ does. We can then define the space C n,d • , λ • associated with d • , λ • , which admits a local description on a dense subspace C ′ n,d • , λ • by § 3.1.2. We can take the 2n|d| = 2n|d • | elements x j , p j , v s,α,j , w s,α,j , j = 1, . . . , n, (s, α) = (0, 1) , (3.17) as coordinates on C ′ n,d • , λ • . Indeed, in view of the constraints (3.7), we can see the (w 0,1,j ) as functions of the variables in (3.17), and the same holds for the (v 0,1,j ) generically.
As in [12, Proposition 6.7], we note that there exists a diffeomorphism 3 Ψ : C n,d • , λ • → C n,d, λ given by We can write the following local expressions on C ′ n,d • , λ • using (3.8) and (3.9) It is clear that the functions (3.19a) with k = 1, . . . , n are functionally independent, since their Jacobian matrix with respect to the coordinates (x 1 , . . . , x n ) is invertible as (x j ) ∈ C n reg . (Without loss of generality, we can replace one of these functions by the Hamiltonian of interest h m,i .) We also note that the functions (3.19b) with k = 1, . . . , n can be used as coordinates instead of (p 1 , . . . , p n ) since the Jacobian matrix with entries is invertible on C ′ n,d • , λ • . It then follows that the functions C m,i k,1 (2.7) with k = 2, . . . , n provide another n − 1 functionally independent first integrals of h m,i due to the identity 21) and the fact that Ψ * tr(Y (i+1)m ) is generically nonzero on C ′ n,d • , λ • . Thus, we have 2n − 1 first integrals of h m,i whose Jacobian matrix taken with respect to the coordinates (q j , p j ) is invertible. We need another 2n(|d|−1) first integrals in order to get the desired dim(C n,d, λ )−1 functionally independent first integrals of h m,i .
Assume that |d| > 1 from now on, otherwise the proof can be concluded here. We will find 2n(|d| − 1) first integrals depending on the coordinates (3.17) with the exception of the (p j ), such that their Jacobian matrix taken with respect to the last 2n(|d| − 1) coordinates in (3.17) is invertible. The functional independence of these new functions and the previous 2n − 1 ones will then follow from this result.
We note that the n first integrals with (s, β) = (0, 1) only depend on the 2n coordinates (q j , v m−s,β,j ). It is straightforward to check that their Jacobian matrix taken with respect to (v m−s,β,j ) is invertible, so that we get a total of n(|d| − 1) additional first integrals which are all functionally independent. There exists s + ∈ {0, . . . , m − 1} such that s + is the maximal index for which d s = 0. Since |d| > 1, the pair (s + , d s+ ) = (0, 1) is such that (v m−s+,ds + ,j ) are n coordinates on C ′ n,d • , λ • from the set (3.17) by construction. Next, we note that the n first integrals Ψ * (t k rα,s+ds + ) = − n j=1 w m−r,α,j v m−s+,ds + ,j x mk+ρr,s + j , k = 1, . . . , n , (3.23) with (r, α) = (0, 1) only depend on the 3n coordinates (q j , w m−r,α,j , v m−s+,ds + ,j ). Since the last n coordinates can be taken to be nonzero at a generic point, we get that the Jacobian matrix is invertible, providing another n(|d| − 1) functionally independent first integrals.
Remark 3.7. We have in fact an explicit integration for the flow of h m,i on the unreduced space Rep(CQ d , n). It follows easily from the following form of the Hamiltonian vector fielḋ This is computed using (3.3).
Remark 3.8. We can easily verify that the functions (h m,i , t i sα,sα ) with 1 ≤ i ≤ n and all possible (s, α) are pairwise Poisson commuting. One can further show that we can form a Liouville integrable system, e.g. by removing the (t i 01,01 ) from these functions and then prove the functional independence of the remaining elements as in Proposition 3.6. This choice of functions is different from the one considered in [12] which is related to the KP hierarchy.

Harmonic CM system
In this section, we fix ω ∈ C × and we consider the Hamiltonian H ω = 1 2 tr(Y 2 + ω 2 X 2 ). We note that it can only be nonzero if m = 1 or m = 2. In those cases, we can remark the following result, see Lemma 5.6 for its proof.

4.1.
Non-spin case. We work over the space C n as in Section 2. We note that on the subset C ′ n we can write , so that we can obtain the CM Hamiltonians with harmonic term of type A n−1 , B n and D n by Remark 2.4. For the root system A n−1 , it was originally introduced by Calogero in the quantum case [7].
is a first integral of H ω .
is a first integral of H ω .
Proof. We apply Theorem 1.1 a) to g j = tr(XY L j + Y XL j ) since {tr L, {tr L, g j }} = −16ω 2 g j by Lemma 5.7. Proof. We first assume that m = 1. We note that as ω → 0, 3) and the latter is just Wojciechowski's function C 2 2k,2j in (2.6). Therefore, the functions tr L, . . . , tr L n , C which can be shown to be independent as in the proof of Proposition 2.3. Thus the functions in (4.4) are independent for generic values of ω.
Remark 4.5. In the real setting, additional first integrals that yield the superintegrability of the harmonic CM system in type A n−1 have been obtained by Adler [1,Section 4]. They are given as the real part of some complex-valued functions, so that we could not directly use them in our setting.

4.2.
Spin case. We work over the space C n,d, λ where n = (n, . . . , n) for some n ∈ N × , as in Section 3.
Example 4.8. When m = 2, the Hamiltonian of interest is H ω = tr(Y 0 Y 1 ) + ω 2 tr(X 0 X 1 ). In the coordinates described in § 3.1.2, we can write In the case d = (d 0 , 0), we get that f

Motivating double brackets.
For researchers in the field of integrable systems, double brackets can be introduced as an analogous approach to finding a Lax matrix and an r-matrix with a different type of derivation rules. To understand this analogy, let us recall that the r-matrix approach can be simplified as finding a matrix L ∈ gl n (C) and an element r ∈ gl n (C) ⊗ gl n (C) such that for a given Poisson bracket, we can write (5.1) Here, L 1 = L ⊗ Id n , L 2 = Id n ⊗L, the left-hand side stands for ijkl {L ij , L kl }E ij ⊗ E kl with E ij the elementary matrix with only nonzero entry equal to +1 in position (i, j), while the permutation operator is defined as The Jacobi identity can also be defined using { {−, −} }, see [40]. Now, an analogue of (5.1) is that if there exist matrices L, (A a ) a∈N such that then the elements (tr L k ) Poisson commute due to the following chain of identities Remark 5.1. There are major differences between the two approaches. First, double Poisson brackets are defined on non-commutative algebras, and their relation to Poisson brackets as explained above is obtained by looking at finite-dimensional representations of the algebras [40]. Second, we have in general that a double Poisson bracket encodes the Poisson bracket on a global phase space, while the tensor notation {− ⊗ , −} is used to understand an associated Poisson bracket obtained in a suitable gauge.

5.2.
Computations on the main space. Fix m ≥ 2, I = Z/mZ, d ∈ N I with d 0 ≥ 1, and n = (n, 1) for n ∈ N I with |n| > 0. We consider the associated quiver Q d and complex Poisson manifold Rep(CQ d , n) as in § 3.1. We can express the Poisson brackets (3.3) in terms of double brackets as (5.11) and it is zero on any other pair of generators (3.1). Note that we see these double brackets as tensor products of square matrices of size |n| + 1, which are elements of End(V) ⊗2 . Introduce from which we note the obvious identities X Id Vs = Id Vs−1 X and Y Id Vs = Id Vs+1 Y in End(V). If we also introduce We are now in position to use double brackets to compute the Poisson brackets between GL(n) invariant functions on Rep(CQ d , n). In particular, these identities descend to the reduced space C n := C n,d, λ . We will repeatedly use (5.5), (5.9b) and (5.10), while we denote Id V as 1 for simplicity.

Computations for Section 3.
Lemma 5.3. Denote by ρ r,s ∈ {0, . . . , m − 1} the representative of s − r ∈ I for any r, s ∈ I. The following identities hold for any indices : Proof. a) Since { {Y, Y } } = 0, the first identity in (5.13a) is obvious. For the second one, we note that b) Since the double brackets of Y with Y, W r,α and V s,β are all zero, this is trivial.
We now compute some Poisson brackets between invariant functions. We restrict to the cases m = 1 and m = 2 as otherwise most of the functions are trivially zero. For example, since L can be decomposed as linear maps V s → V s±2 for all s ∈ I, tr L can only be nonzero in those two cases. If m = 1, 2, we have for any l ∈ N, {tr L, tr XY L l } ={tr L, tr Y XL l } = −2 tr(Y 2 L l ) + 2ω 2 tr(X 2 L l ) , (5.19a) {tr L, tr Y 2 L l } =2ω 2 tr(XY L l ) + tr(Y XL l ) , (5.19b) {tr L, tr X 2 L l } = − 2 tr(XY L l ) + tr(Y XL l ) .

Conclusion and outlook
In this paper, we focused on establishing superintegrability of complex generalisations of the rational CM system associated with cyclic quivers. These various systems are allowed to admit different types of spin variables (internal degrees of freedom) or a harmonic oscillator potential term, which are completely determined by the underlying quivers.
To continue the investigation reported in this paper, it seems natural to try to construct such generalisations for other systems in the Calogero-Ruijsenaars family of integrable n-particle systems. In fact, these generalisations for the trigonometric RS system are known: for the simplest quivers considered in Section 2, the corresponding systems were constructed in [10]; for the general quivers considered in Section 3, the systems can be found in [11,16,17]. Our next aim is to unveil generalisations of the trigonometric CM and rational RS systems associated with cyclic quivers, which we expect to be maximally superintegrable. While the quivers from Section 2 give the usual form of these systems (see [25, §4.5]), the quivers from Section 3 lead to new versions of these systems endowed with different types of spin variables. In particular, we will investigate if they can be connected with the spin systems studied in [13,20,21,27].