Integrable system of generalized relativistic interacting tops

Abstract

We describe a family of integrable $GL(NM)$ models generalizing classical spin Ruijsenaars–Schneider systems (the case $N=1$) on one hand and relativistic integrable tops on the $GL(N)$ Lie group (the case $M=1$) on the other hand. We obtain the described models using the Lax pair with a spectral parameter and derive the equations of motion. To construct the Lax representation, we use the $GL(N)$ $R$-matrix in the fundamental representation of $GL(N)$.

1. Introduction

This paper continues a series of articles [1]–[3] extending known integrable systems and their related structures using quantum $R$-matrices (in the fundamental representation of $GL(N)$ Lie groups) interpreted as matrix generalizations of the Kronecker function. At the very beginning, it is convenient to give its explicit form in the rational, trigonometric, and elliptic cases because the identities that we use hold separately in each of these cases:

$$\phi(z,q)=\begin{cases}\dfrac{1}{z}+\dfrac{1}{q},\\ \coth z+\coth q,\\\dfrac{ \vartheta '(0) \vartheta (z+q)}{ \vartheta (z) \vartheta (q)}. \end{cases}\quad E_1(z)=\begin{cases}\dfrac{1}{z},\\\coth(z),\\ \dfrac{ \vartheta '(z)}{ \vartheta (z)}\end{cases}\quad \wp(z)=\begin{cases}\dfrac{1}{z^2},\\\dfrac{1}{\sinh^2 z},\\ -E_1'(z)+\dfrac{1}{3}\dfrac{ \vartheta '''(0)}{ \vartheta '(0)}.\end{cases} $$

(1)

All the variables and functions are complex valued. Therefore, the trigonometric and hyperbolic cases are essentially the same. In (1) in all three cases, we give definitions of the first Eisenstein function $E_1(z)$ and the Weierstrass $\wp$-function. They appear in the expansion of $\phi(z,q)$ near its simple pole (with the residue equal to unity) at $z=0$:

$$\phi(z,q)=z^{-1}+E_1(q)+\frac{z(E_1^2(q)-\wp(q))}{2}+O(z^2). $$

(2)

The definitions and properties of elliptic functions can be found in [4] (also see the appendix in [3]).

In each of the three cases, the Kronecker function satisfies the summation formula, the genus-1 Fay identity,

$$\phi(z_1,q_1)\phi(z_2,q_2)=\phi(z_1-z_2,q_1)\phi(z_2,q_1+q_2)+ \phi(z_2-z_1,q_2)\phi(z_1,q_1+q_2), $$

(3)

and also its degenerations corresponding to equal arguments,

$$\phi(z,q_1)\phi(z,q_2)=\phi(z,q_1+q_2)(E_1(z)+ E_1(q_1)+E_1(q_2)-E_1(q_1+q_2+z)),$$

(4)

$$\phi(z,q)\phi(z,-q)=\wp(z)-\wp(q).$$

(5)

Fay identity (3) can be regarded as a particular scalar case of the associative Yang–Baxter equation [5]:

$$R^z_{12}(q_{12})R^{w}_{23}(q_{23})=R^{w}_{13}(q_{13})R_{12}^{z-w}(q_{12})+ R^{w-z}_{23}(q_{23})R^z_{13}(q_{13}),\qquad q_{ab}=q_a-q_b. $$

(6)

Here, we use $R$-matrix notation of the quantum inverse scattering method, for example,

$$\begin{aligned} \, &R^z_{12}(q)=\sum_{i,j,k,l=1}^NR_{ij,kl}(z,q)E_{ij}\otimes E_{kl}\otimes1_N, \\ &R^z_{13}(q)=\sum_{i,j,k,l=1}^NR_{ij,kl}(z,q)E_{ij}\otimes1_N\otimes E_{kl}, \end{aligned} $$

(7)

where $E_{ij}$ is the standard matrix basis in $\operatorname{Mat}(N,\mathbb{C})$, $1_N$ is the identity matrix, and $R_{ij,kl}(z,q)$ is a set of functions of $z$ and $q$. The normalization of the matrix operator $R^z_{ab}(q_{ab})$ is chosen such that with $N=1$, it reduces to the scalar function $\phi(z,q)$ given by (1). In this respect, Eq. (6) is a noncommutative generalization of (3), and the operator $R$ is a noncommutative generalization of the Kronecker function.

In addition to (6), we can require the properties of antisymmetry and unitarity (the latter is a matrix analogue of (5)):

$$R^z_{12}(q)=-R^{-z}_{21}(-q),\qquad R^z_{12}(q)R^z_{21}(-q)=1_N\otimes1_N(\wp(z)-\wp(q)). $$

(8)

Then such an $R$-operator satisfies the quantum Yang–Baxter equation

$$R_{12}^\hbar(z_{12})R_{13}^\hbar(z_{13})R_{23}^\hbar(z_{23})= R_{23}^\hbar(z_{23})R_{13}^\hbar(z_{13})R_{12}^\hbar(z_{12}). $$

(9)

In other words, a solution of (6) satisfying conditions (8) is a quantum $R$-matrix. We note that even in the scalar case, condition (6) or (3) is very restrictive. At the same time, Eq. (9) is not restrictive at all because the quantum Yang–Baxter equation holds identically in the scalar case. The class of $R$-matrices with the listed properties includes the elliptic Baxter–Belavin $R$-matrix and also its trigonometric and rational degenerations, which are equal to the function $\phi(z,q)$ in the scalar case. A more detailed description of these $R$-matrices can be found in [6]–[8], where an application of this class of $R$-matrices to an integrable system was given, a construction of integrable tops. The main idea goes back to Sklyanin’s paper [9], where he suggested a Hamiltonian description of the classical Euler top using quadratic Poisson algebras obtained in the classical limit of $RLL$ relations. That is, the classical Euler top was described as the classical limit of a spin chain with one site. This approach can be developed to obtain an explicit description of the Lax pairs with spectral parameters constructed using the data of $R$-matrices satisfying (6) and (8). A detailed derivation of the equations of motion together with the Hamiltonian description using the $R$-matrix data was given in [6] and [7] in the respective nonrelativistic and relativistic cases.

1.1. Relativistic integrable $GL_N$-top

In the general case, the phase space of a $GL_N$ top is given by the set of coordinate functions $S_{ij}$, $i,j=1,\dots,N$, on the Lie group $GL_N$. They are unified into the $N{\times}N$ matrix $S=\sum_{ij}S_{ij}E_{ij}$. The equations of motion then become the Euler–Arnold equations

$$\dot {S}=[S,J(S)], $$

(10)

where $J(S)$ is a linear functional on $S$. It can be written as

$$J(S)=\sum_{i,j,k,l=1}^NJ_{i,j,k,l}\,E_{ij}\,S_{lk}\in \operatorname{Mat}(N,\mathbb{C}) $$

(11)

or, using the standard notation $S_1=S\otimes1_N$ and $S_2=1_N\otimes S$,

$$J(S)= \operatorname{tr} _2(J_{12}S_2),\qquad J_{12}=\sum_{i,j,k,l=1}^NJ_{i,j,k,l}\,E_{ij}\otimes E_{kl}, $$

(12)

where $\operatorname{tr} _2$ is the trace over the second space in the tensor product. Below, we give the Lax pair of the relativistic integrable top using the above notation (of course, Eq. (10) is not integrable in the general case). For this, we consider the classical limit of the $R$-matrix

$$R^\hbar_{12}(z)=\frac{1}{\hbar}\,1_N\otimes1_N+r_{12}(z)+ \frac{\hbar}{2}\,(r_{12}(z)^2-1\otimes1\,\wp(z))+O(\hbar^2), $$

(13)

where $r_{12}(z)=-r_{21}(-z)$ is the classical $r$-matrix and the $\hbar$-order term follows from (8). Comparing this expression with (2), we conclude that while the quantum $R$-matrix is a matrix analogue of the Kronecker function, the classical $r$-matrix is a matrix analogue of the first Eisenstein function $E_1(z)$ given in (1).

We consider the expansions

$$\begin{aligned} \, &R^{\,z}_{12}(q)=\frac{1}{q}P_{12}+R^{z,(0)}_{12}+O(q), \qquad r_{12}(z)=\frac{1}{z}P_{12}+r^{(0)}_{12}+O(z), \\ &P_{12}=\sum_{i,j=1}^NE_{ij}\otimes E_{ji}, \end{aligned} $$

(14)

where $P_{12}$ is the matrix permutation operator. Generally speaking, the existence of expansions of types (13) and (14) is an additional nontrivial requirement for the $R$-matrix. Finally, we impose one more condition on the $R$-matrix:

$$R^{\,z}_{12}(q)=R^{\,q}_{12}(z)P_{12}. $$

(15)

In the scalar case, it leads to the obvious equality $\phi(z,q)=\phi(q,z)$. Using (15) and comparing (13) and (14), we easily obtain

$$r_{12}(z)=R^{z,(0)}_{12}P_{12}. $$

(16)

We can now formulate the statement about the Lax pair of the relativistic top. Namely, for a pair of matrices

$$\begin{aligned} \, &L(z)= \operatorname{tr} _2(R^\eta_{12}(z)S_2)= \operatorname{tr} _2(R^{\,z}_{12}(\eta)P_{12}S_2), \\ &M(z)=- \operatorname{tr} _2(r_{12}(z)S_2)=- \operatorname{tr} _2(R^{z,(0)}_{12}P_{12}S_2), \end{aligned} $$

(17)

the Lax equation

$$\dot{L}(z)=[L(z),M(z)] $$

(18)

is equivalent to the equation of motion of form (10), where

$$J_{12}=R^{\eta,(0)}_{12}-r^{(0)}_{12}. $$

(19)

1.2. Spin generalization of the Ruijsenaars–Schneider model

In integrable many-body systems, relativistic generalizations are known as Ruijsenaars–Schneider models [10]. We are interested in their spin extensions. The dynamical variables comprise a set of coordinates and velocities of $M$ particles and also classical spin variables described in the matrix $S\in \operatorname{Mat}(M,\mathbb{C})$. The equations of motion have the forms (for the diagonal and off-diagonal parts of $S$)

$$\begin{aligned} \, &{\dot S}_{ii}=-\sum_{k\colon k\ne i}^MS_{ik}S_{ki}(E_1(q_{ik}+\eta)+ E_1(q_{ik}-\eta)-2E_1(q_{ik})), \\ &{\dot S}_{ij}=\sum_{k\colon k\ne j}^MS_{ik}S_{kj}( E_1(q_{kj}+ \eta)-E_1(q_{kj}))-\sum_{k\colon k\ne i}^M S_{ik}S_{kj}(E_1(q_{ik}+\eta)-E_1(q_{ik})) \end{aligned} $$

(20)

and

$${\ddot q}_i={\dot S}_{ii}, $$

(21)

where $i\ne j$ and $q_{ij}=q_i-q_j$. The Lax pair with a spectral parameter

$$\begin{aligned} \, &L_{ij}(z)=S_{ij}\phi(z,q_{ij}+\eta),\quad i,j=1,\dots,M, \\ & \mathop{\operatorname{Res}} _{z=0}L(z)=S\in \operatorname{Mat}(M,\mathbb{C}) , \\ &M_{ij}(z)=-\delta_{ij}(E_1(z)+E_1(\eta))S_{ii}- (1-\delta_{ij})S_{ij}\phi(z,q_{ij}), \end{aligned} $$

(22)

satisfies the Lax equation with an additional term (here $\mu_i=\dot q_i-S_{ii}$)

$$\begin{aligned} \, \dot{L}(z)=[L(z),M(z)]+\sum_{i,j=1}^ME_{ij}(\mu_i-\mu_j)S_{ij} f(z,q_{ij}+\eta),\qquad f(z,q)= \partial _q\phi(z,q), \end{aligned} $$

(23)

which vanishes with the on-shell constraints

$$\mu_i=0\qquad\text{or}\qquad S_{ii}=\dot q_i,\quad i=1,\dots,M. $$

(24)

More precisely, Eq. (23) is equivalent to (20), and under condition (24), Lax equations (23) with an additional term become the ordinary Lax equations (18), and (21) is satisfied. A detailed derivation of (23) can be found in [3] in addition to the original paper [11]. This derivation is convenient below in considering a more general system where the functions in (20)–(22) are replaced with their $R$-matrix analogues. Although the Hamiltonian structure is not used in the description indicated above, we note that it is known for the rational and trigonometric systems (see [12]–[15]).

Our main result here is the following generalization of simultaneously both relativistic top (17)–(19) and spin Ruijsenaars–Schneider model (20)–(22). We consider a $\operatorname{Mat}(N\!M,\mathbb{C})$-valued Lax pair subdivided into $M{\times}M$ block-matrices $\mathcal{L} ^{ij}(z)= \mathcal{L} ^{ij}( \mathcal{S} ^{ij},z)$ each of size $N\times N$:

$$\begin{aligned} \, & \mathcal{L} (z)=\sum_{i,j=1}^ME_{ij}\otimes \mathcal{L} ^{ij}(z)\in \operatorname{Mat}(N\!M,\mathbb{C}) ,\quad \mathcal{L} ^{ij}(z)\in \operatorname{Mat}(N,\mathbb{C}) , \\ & \mathcal{L} ^{ij}(z)= \operatorname{tr} _2 (R_{12}^z(q_{ij}+\eta)P_{12} \mathcal{S} ^{ij}_2),\quad \mathcal{S} ^{ij}= \mathop{\operatorname{Res}} _{z=0} \mathcal{L} ^{ij}(z)\in \operatorname{Mat}(N,\mathbb{C}) , \\ & \mathcal{M} (z)=\sum_{i,j=1}^M E_{ij}\otimes \mathcal{M} ^{ij}(z)\in \operatorname{Mat}(N\!M,\mathbb{C}) ,\quad \mathcal{M} ^{ij}(z)\in \operatorname{Mat}(N,\mathbb{C}) , \\ & \mathcal{M} ^{ij}(z)=-\delta^{ij}\, \operatorname{tr} _2\bigl(R_{12}^{(0),z}P_{12} \mathcal{S} ^{ii}_2\bigr)- (1-\delta^{ij})\, \operatorname{tr} _2\bigl(R_{12}^z(q_{ij}) P_{12} \mathcal{S} ^{ij}_2\bigr). \end{aligned} $$

(25)

The $R$-matrix in this definition satisfies associative Yang–Baxter equation (6) and also conditions (8) and (15) and expansions (13) and (14). The Lax equation with an additional term

$$\dot{ \mathcal{L} }(z)=[ \mathcal{L} (z), \mathcal{M} (z)]+\sum_{i,j=1}^M(\mu^i_0-\mu^j_0)\, E_{ij}\otimes \operatorname{tr} _2(F_{12}^z(q_{ij}+\eta)P_{12} \mathcal{S} ^{ij}_2), $$

(26)

where by analogy with (23)

$$F_{12}^z(q)= \partial _qR_{12}^z(q) $$

(27)

and $\mu^i_0={\dot q}_i- \operatorname{tr} ( \mathcal{S} ^{ii})$, $i=1,\dots,M$, is then equivalent to the equations of motion (we assume that $i\ne j$ in (26))

$${\dot{ \mathcal{S} }}^{ii}=[ \mathcal{S} ^{ii},J^\eta( \mathcal{S} ^{ii})]+\sum_{k\colon k\ne i}^M \bigl( \mathcal{S} ^{ik}J^{\eta,q_{ki}}(S^{ki})-J^{\eta,q_{ik}}( \mathcal{S} ^{ik})S^{ki}\bigr),$$

(28)

$${\dot{ \mathcal{S} }}^{ij}= \mathcal{S} ^{ij}J^\eta( \mathcal{S} ^{jj})-J^\eta( \mathcal{S} ^{ii}) \mathcal{S} ^{ij}+ \sum_{k\colon k\ne j}^M \mathcal{S} ^{ik}J^{\eta,q_{kj}}( \mathcal{S} ^{kj})- \sum_{k\colon k\ne i}^MJ^{\eta,q_{ik}}( \mathcal{S} ^{ik}) \mathcal{S} ^{kj}.$$

(29)

With the on-shell constraints $\mu^i_0=0$ or $\dot q_i= \operatorname{tr} ( \mathcal{S} ^{ii})$, $i=1,\dots,M$, Eqs. (26) reduce to the Lax equations, and we have the equations

$$\ddot q_i= \operatorname{tr} (\dot{ \mathcal{S} }^{ii})=\sum_{k\colon k\ne i}^M \operatorname{tr} \bigl( \mathcal{S} ^{ik}J^{\eta,q_{ki}}( \mathcal{S} ^{ki})- J^{\eta,q_{ik}}( \mathcal{S} ^{ik}) \mathcal{S} ^{ki}\bigr). $$

(30)

The linear functionals $J^\eta$ and $J^{\eta, q}$ in the equations of motion are given by

$$J^\eta( \mathcal{S} ^{ii})= \operatorname{tr} _2\bigl((R_{12}^{(0),\eta}- r_{12}^{(0)}) \mathcal{S} _2^{ii}\bigr),\qquad J^{\eta,q}( \mathcal{S} ^{ij})= \operatorname{tr} _2\bigl((R_{12}^{(0),q+\eta}-R_{12}^{(0),q}) \mathcal{S} _2^{ij}\bigr). $$

(31)

The presented Lax pairs and equations of motion reproduce the results in the elliptic case in our previous paper [3]¹ and the results in the nonrelativistic limit in [1]. With $N=1$, the used $R$-matrix operators become the scalar functions in (1), thus reproducing spin Ruijsenaars–Schneider model (20)–(24). With $M=1$, the Lax matrices have a single block. We thus obtain relativistic top (17)–(19). In the nonrelativistic elliptic case, models of the described type were first obtained in [16] and were later described as Hitchin systems on bundles with nontrivial characteristic classes [17]. Explicit examples of the systems can be easily obtained using $R$-matrices used in [1] in the same normalization as here.

2. Derivation of the equations of motion

2.1. $R$-matrix identities

To derive the equations of motion in the spin Ruijsenaars–Schneider model, we should use identity (4). We rewrite it differently:

$$\phi(z,q_1)\phi(z,q_2)= \phi(z,q_1+q_2)(E_1(q_1)+E_1(q_2))- \partial _z \phi(z,q_1+q_2), $$

(32)

where we use the fact that (1) implies that $\partial _z \phi(z,q)= \phi(z,q)(E_1(z+q)-E_1(z))$. In this form, identity (4) is generalized to the matrix case:

$$R_{12}^z(x) R_{23}^z(y)=R_{13}^z(x+y)r_{12}(x)+ r_{23}(y)R_{13}^z(x+y)-\frac{ \partial }{ \partial z}R_{13}^z(x+y). $$

(33)

Applications of this identity can be found in [8]. We also write its corollary:

$$\begin{aligned} \, R_{12}^z(q_{ik})&R_{23}^z(q_{kj}+\eta)- R_{12}^z(q_{ik}+\eta)R_{23}^z(q_{kj})= \nonumber \\ &=R_{13}^z(q_{ij}+\eta)(r_{12}(q_{ik})-r_{12}(q_{ik}+\eta))+ (r_{23}(q_{kj}+\eta)-r_{23}(q_{kj}))R_{13}^z(q_{ij}+\eta). \end{aligned}$$

(34)

Moreover, we need degenerations of (33). We expand both its sides in a neighborhood of $x=0$:

$$\begin{aligned} \, \biggl(\frac{1}{x}P_{12}+R_{12}^{(0),z}+\cdots\biggr)R_{23}^z(y)={}& \bigl(R_{13}^z(y)+xF_{13}^z(y)+\cdots\bigr) \biggl(\frac{1}{x}P_{12}+r_{12}^{(0)}+\cdots\biggr)+{} \nonumber \\ &{}+r_{23}(y)\bigl(R_{13}^z(y)+xF_{13}^z(y)+\cdots\bigr)- \frac{ \partial }{ \partial z}\bigl(R_{13}^z(y)+x F_{13}^z(y)+\dots\bigr), \end{aligned}$$

(35)

where $F_{ab}^z(y)$ is defined as in (27). In the zeroth order in $x$, we obtain

$$R_{12}^{(0),z}R_{23}^z(y)=F_{13}^z(y)P_{12}+R_{13}^z(y)r_{12}^{(0)}+ r_{23}(y)R_{13}^z(y)-\frac{ \partial }{ \partial z}R_{13}^z(y). $$

(36)

Expanding (33) with small $y$, we similarly obtain

$$\begin{aligned} \, R_{12}^z(x)\biggl(\frac{1}{y}P_{23}+R_{23}^{(0),z}+\cdots\biggr)={}& \bigl(R_{13}^z(x)+yF_{13}^z(x)+\cdots\bigr)r_{12}(x)+{} \nonumber \\ &{}+\biggl(\frac{1}{y}P_{23}+r_{23}^{(0)}+\cdots\biggr) \bigl(R_{13}^z(x)+yF_{13}^z(x)+\cdots\bigr)-{} \nonumber \\ &{}-\frac{ \partial }{ \partial z}\bigl(R_{13}^z(x)+yF_{13}^z(x)+\dots\bigr), \end{aligned}$$

(37)

$$R_{12}^z(x)R_{23}^{(0),z}=R_{13}^z(x)r_{12}(x)+r_{23}^{(0)}R_{13}^z(x)+ P_{23}F_{13}^z(x)-\frac{ \partial }{ \partial z}R_{13}^z(x). $$

(38)

It follows from (36) and (38) that

$$\begin{aligned} \, R_{12}^{(0),z}R_{23}^z(q_{ij}+\eta)-R_{12}^z(\eta)R_{23}^z(q_{ij})={}& F_{13}^z(q_{ij}+\eta)P_{12}+ R_{13}^z(q_{ij}+\eta)(r_{12}^{(0)}-r_{12}(\eta))+{} \nonumber \\ &{}+(r_{23}(q_{ij}+\eta)-r_{23}(q_{ij}))R_{13}^z(q_{ij}+\eta). \end{aligned}$$

(39)

2.2. Lax equation

We write Lax equation (26) with an additional term explicitly in terms of $N{\times}N$ blocks. For the diagonal blocks, we obtain

$$\dot{ \mathcal{L} }_{ii}(z)= \mathcal{L} ^{ii}(z) \mathcal{M} ^{ii}(z)- \mathcal{M} ^{ii}(z) \mathcal{L} ^{ii}(z)+ \sum_{k \ne i}\bigl( \mathcal{L} ^{ik}(z) \mathcal{M} ^{ki}(z)- \mathcal{M} ^{ik}(z) \mathcal{L} ^{ki}(z)\bigr). $$

(40)

Similarly, for the off-diagonal part, we have

$$\begin{aligned} \, \dot{ \mathcal{L} }_{ij}(z)={}& \mathcal{L} ^{ii}(z) \mathcal{M} ^{ij}(z)- \mathcal{M} ^{ii}(z) \mathcal{L} ^{ij}(z)+ \mathcal{L} ^{ij}(z) \mathcal{M} ^{jj}(z)- \mathcal{M} ^{ij}(z) \mathcal{L} ^{jj}(z)+{} \nonumber \\ &{}+\sum_{k \ne i,j}\bigl( \mathcal{L} ^{ik}(z) \mathcal{M} ^{kj}(z)- \mathcal{M} ^{ik}(z) \mathcal{L} ^{kj}(z)\bigr)+{} \nonumber \\ &{}+(\mu^i_0-\mu^j_0)\, \operatorname{tr} _2\bigl(F_{12}^z(q_{ij}+ \eta)P_{12} \mathcal{S} ^{ij}_2\bigr). \end{aligned}$$

(41)

The problem is to show that (40) and (41) are equivalent to the respective equations of motion (28) and (29). We note that

$$\mathop{\operatorname{Res}} _{z=0} \mathcal{L} (z)= S=- \mathop{\operatorname{Res}} _{z=0} \mathcal{M} (z)\in \operatorname{Mat}(N\!M,\mathbb{C}) , $$

(42)

i.e., the second-order pole in $z$ cancels in the commutator $[ \mathcal{L} (z), \mathcal{M} (z)]$.

2.2.1. Off-diagonal part

In the left-hand side of (41), we have

$$\dot{ \mathcal{L} }^{ij}(z)_1= \operatorname{tr} _2(F_{12}^z(q_{ij}+\eta)P_{12} \mathcal{S} ^{ij}_2 \dot{q}_{ij})+ \operatorname{tr} _2(R_{12}^z(q_{ij}+\eta) P_{12}\dot{ \mathcal{S} }^{ij}_2). $$

(43)

The subscript 1 in the left-hand side means that the Lax equation is in the first tensor component. We consider expression in the right-hand side of (41):

$$\begin{aligned} (\mathcal{L} ^{ik}(z) \mathcal{M} ^{kj}(z)&{}- \mathcal{M} ^{ik}(z) \mathcal{L} ^{kj}(z))_1= \\ &\;= \operatorname{tr} _{23}\bigl(-R_{12}^z(q_{ik}+\eta)P_{12} \mathcal{S} _2^{ik} R_{13}^z(q_{kj})P_{13} \mathcal{S} _3^{kj}+{} \\ &\quad\;\,+R_{12}^z(q_{ik})P_{12} \mathcal{S} _2^{ik} R_{13}^z(q_{kj}+\eta)P_{13} \mathcal{S} _3^{kj}\bigr)= \\ &\;= \operatorname{tr} _{23}\bigl((R_{12}^z(q_{ik})R_{23}^z(q_{kj}+ \eta)-R_{12}^z(q_{ik}+\eta)R_{23}^z(q_{kj})) P_{12} \mathcal{S} _2^{ik}P_{13} \mathcal{S} _3^{kj}\bigr)\stackrel{(34)}{=} \\ &\stackrel{(34)}{=} \operatorname{tr} _{23}\bigl(R_{13}^z(q_{ij}+\eta)(r_{12}(q_{ik})- r_{12}(q_{ik}+\eta))P_{12} \mathcal{S} _2^{ik}P_{13} \mathcal{S} _3^{kj}\bigr)+{} \\ &\qquad+ \operatorname{tr} _{23}\bigl((r_{23}(q_{kj}+\eta)-r_{23}(q_{kj})) R_{13}^z(q_{ij}+\eta)P_{12} \mathcal{S} _2^{ik}P_{13} \mathcal{S} _3^{kj}\bigr). \end{aligned} $$

(44)

We transform the two obtained terms using (15) and (16) and the respective permutation operator properties $P_{12}U_{12}= U_{21}P_{12}$ and $P_{12}U_{23}= U_{13}P_{12}$. We transform the first term in the right-hand side of (44):

$$\begin{aligned} \operatorname{tr} _{23}\bigl(R_{13}^z(q_{ij}+\eta)(r_{12}(q_{ik})&{}- r_{12}(q_{ik}+\eta))P_{12} \mathcal{S} _2^{ik}P_{13} \mathcal{S} _3^{kj}\bigr)= \\ &=- \operatorname{tr} _3\bigl(R_{13}^z(q_{ij}+\eta)P_{13}P_{13}\, \operatorname{tr} _2((R_{12}^{(0),q_{ik}+ \eta}-R_{12}^{(0),q_{ik}}) \mathcal{S} _2^{ik})P_{13} \mathcal{S} _3^{kj}\bigr)= \\ &=- \operatorname{tr} _3\bigl(R_{13}^z(q_{ij}+\eta)P_{13}\, \operatorname{tr} _2((R_{32}^{(0),q_{ik}+ \eta}-R_{32}^{(0),q_{ik}}) \mathcal{S} _2^{ik}) \mathcal{S} _3^{kj}\bigr)= \\ &=- \operatorname{tr} _2\bigl(R_{12}^z(q_{ij}+\eta)P_{12}\, \operatorname{tr} _3((R_{23}^{(0),q_{ik}+ \eta}-R_{23}^{(0),q_{ik}}) \mathcal{S} _3^{ik}) \mathcal{S} _2^{kj}\bigr). \end{aligned} $$

(45)

Using definition (31), we obtain

$$\operatorname{tr} _{23} \bigl(R_{13}^z(q_{ij}+\eta) (r_{12}(q_{ik})-r_{12}(q_{ik}+\eta)) P_{12} \mathcal{S} _2^{ik} P_{13} \mathcal{S} _3^{kj}\bigr) = - \operatorname{tr} _2 \bigl(R_{12}^z(q_{ij}+\eta) P_{12} {J^{\eta, q_{ik}}( \mathcal{S} ^{ik})}_2\, \mathcal{S} _2^{kj}\bigr). $$

(46)

We similarly transform the second term in the right-hand side of (44):

$$\begin{aligned} \operatorname{tr} _{23}\bigl((r_{23}(q_{kj}+\eta)&{}-r_{23}(q_{kj}))R_{13}^z(q_{ij}+\eta) P_{12} \mathcal{S} _2^{ik}P_{13} \mathcal{S} _3^{kj}\bigr)= \\ &= \operatorname{tr} _{23}\bigl(R_{13}^z(q_{ij}+\eta)P_{12} \mathcal{S} _2^{ik}P_{13} \mathcal{S} _3^{kj} (r_{23}(q_{kj}+\eta)-r_{23}(q_{kj}))\bigr)= \\ &= \operatorname{tr} _{23}\bigl(R_{13}^z(q_{ij}+\eta)P_{13}P_{23} \mathcal{S} _2^{ik} \mathcal{S} _3^{kj} (R_{23}^{(0),q_{kj}+\eta}-R_{23}^{(0),q_{kj}})P_{23}\bigr)= \\ &= \operatorname{tr} _{23}\bigl(P_{23}R_{13}^z(q_{ij}+\eta)P_{13}P_{23} \mathcal{S} _2^{ik} \mathcal{S} _3^{kj} (R_{23}^{(0),q_{kj}+\eta}-R_{23}^{(0),q_{kj}})\bigr)= \\ &= \operatorname{tr} _{23}\bigl(R_{12}^z(q_{ij}+\eta)P_{12} \mathcal{S} _2^{ik} \mathcal{S} _3^{kj} (R_{23}^{(0),q_{kj}+\eta}-R_{23}^{(0),q_{kj}})\bigr)= \\ &= \operatorname{tr} _2\bigl(R_{12}^z(q_{ij}+\eta)P_{12} \mathcal{S} _2^{ik} \, \operatorname{tr} _3\,( \mathcal{S} _3^{kj}(R_{23}^{(0),q_{kj}+\eta}-R_{23}^{(0),q_{kj}}))\bigr)= \\ &= \operatorname{tr} _2\bigl(R_{12}^z(q_{ij}+\eta)P_{12} \mathcal{S} _2^{ik}J^{\eta,q_{kj}}( \mathcal{S} ^{kj})_2\bigr). \end{aligned} $$

(47)

From (46) and (47), for the initial expression (44), we finally obtain

$$\bigl( \mathcal{L} ^{ik}(z) \mathcal{M} ^{kj}(z)- \mathcal{M} ^{ik}(z) \mathcal{L} ^{kj}(z)\bigr)_1 = \operatorname{tr} _2 \bigl(R_{12}^z(q_{ij}+\eta) P_{12} ( \mathcal{S} ^{ik} J^{\eta, q_{kj}}( \mathcal{S} ^{kj})-J^{\eta, q_{ik}}( \mathcal{S} ^{ik}) \mathcal{S} ^{kj})_2 \bigr). $$

(48)

We next consider the expression from (41)

$$\begin{aligned} \bigl( \mathcal{L} ^{ii}(z) \mathcal{M} ^{ij}(z)&{}- \mathcal{M} ^{ii}(z) \mathcal{L} ^{ij}(z)\bigr)_1= \\ &= \operatorname{tr} _{23}\bigl(-R_{12}^z(\eta)P_{12} \mathcal{S} _2^{ii}R_{13}^z(q_{ij}) P_{13} \mathcal{S} _3^{ij}+R_{12}^{(0),z}P_{12} \mathcal{S} _2^{ii}R_{13}^z(q_{ij}+ \eta)P_{13} \mathcal{S} _3^{ij}\bigr)= \\ &= \operatorname{tr} _{23}\bigl((R_{12}^{(0),z}R_{23}^z(q_{ij}+\eta)-R_{12}^z(\eta) R_{23}^z(q_{ij}))P_{12} \mathcal{S} _2^{ii}P_{13} \mathcal{S} _3^{ij}\bigr). \end{aligned} $$

(49)

We apply relation (39),

$$\begin{aligned} \bigl( \mathcal{L} ^{ii}(z) \mathcal{M} ^{ij}(z)- \mathcal{M} ^{ii}(z) \mathcal{L} ^{ij}(z)\bigr)_1={}& \operatorname{tr} _{23}\bigl(F_{13}^z(q_{ij}+\eta) P_{12}P_{12} \mathcal{S} _2^{ii}P_{13} \mathcal{S} _3^{ij}\bigr)+{} \\ &{}+ \operatorname{tr} _{23}\bigl(R_{13}^z(q_{ij}+\eta) (r_{12}^{(0)}-r_{12}(\eta))P_{12} \mathcal{S} _2^{ii}P_{13} \mathcal{S} _3^{ij}\bigr)+{} \\ &{}+ \operatorname{tr} _{23}\bigl((r_{23}(q_{ij}+\eta)-r_{23}(q_{ij})) R_{13}^z(q_{ij}+\eta)P_{12} \mathcal{S} _2^{ii}P_{13} \mathcal{S} _3^{ij}\bigr). \end{aligned} $$

(50)

We simplify all three terms in the right-hand side of (50). We transform the first term,

$$\begin{aligned} \operatorname{tr} _{23}\bigl(F_{13}^z(q_{ij}+\eta)P_{12}P_{12} \mathcal{S} _2^{ii}P_{13} \mathcal{S} _3^{ij}\bigr)&= \operatorname{tr} _2( \mathcal{S} _2^{ii})\, \operatorname{tr} _3\bigl(F_{13}^z(q_{ij}+\eta)P_{13} \mathcal{S} _3^{ij}\bigr)= \\ &= \operatorname{tr} \mathcal{S} ^{ii}\cdot \operatorname{tr} _2\bigl(F_{12}^z(q_{ij}+\eta)P_{12} \mathcal{S} _2^{ij}\bigr). \end{aligned} $$

(51)

The third term is already known:

$$\operatorname{tr} _{23}\bigl((r_{23}(q_{ij}+\eta)-r_{23}(q_{ij}))R_{13}^z(q_{ij}+\eta) P_{12} \mathcal{S} _2^{ii}P_{13} \mathcal{S} _3^{ij}\bigr)= \operatorname{tr} _2\bigl(R_{12}^z(q_{ij}+\eta) P_{12} \mathcal{S} _2^{ii}J^{\eta,q_{ij}}( \mathcal{S} ^{ij})_2\bigr). $$

(52)

For the second term in the right-hand side of (50), we obtain

$$\begin{aligned} \operatorname{tr} _{23}\bigl(R_{13}^z(q_{ij}+\eta)(r_{12}^{(0)}&{}- r_{12}(\eta))P_{12} \mathcal{S} _2^{ii}P_{13} \mathcal{S} _3^{ij}\bigr)= \\ &= \operatorname{tr} _{23}\bigl(R_{13}^z(q_{ij}+\eta)P_{13} (r_{32}^{(0)}-r_{32}(\eta))P_{32} \mathcal{S} _2^{ii} \mathcal{S} _3^{ij}\bigr)= \\ &= \operatorname{tr} _2\bigl(R_{12}^z(q_{ij}+\eta)P_{12}\, \operatorname{tr} _3\bigl((r_{23}^{(0)}-r_{23} (\eta))P_{23} \mathcal{S} _3^{ii}\bigr) \mathcal{S} _2^{ij}\bigr)= \\ &=- \operatorname{tr} _2\bigl(R_{12}^z(q_{ij}+\eta)P_{12}J^\eta( \mathcal{S} ^{ii})_2 \mathcal{S} _2^{ij}\bigr). \end{aligned} $$

(53)

Expression (50) thus becomes

$$\begin{aligned} \bigl( \mathcal{L} ^{ii}(z) \mathcal{M} ^{ij}(z)- \mathcal{M} ^{ii}(z) \mathcal{L} ^{ij}(z)\bigr)_1={}& \operatorname{tr} _2\bigl(R_{12}^z(q_{ij}+\eta)P_{12} \bigl( \mathcal{S} ^{ii}J^{\eta,q_{ij}}( \mathcal{S} ^{ij})-J^\eta( \mathcal{S} ^{ii}) \mathcal{S} ^{ij})_2\bigr)+{} \\ &{}+ \operatorname{tr} \mathcal{S} ^{ii}\cdot \operatorname{tr} _2\bigl(F_{12}^z(q_{ij}+\eta)P_{12} \mathcal{S} _2^{ij}\bigr). \end{aligned} $$

(54)

We transform one more expression in (41), $\mathcal{L} ^{ij}(z) \mathcal{M} ^{jj}(z)- \mathcal{M} ^{ij}(z) \mathcal{L} ^{jj}(z)$, similarly to (50). This yields

$$\begin{aligned} \bigl( \mathcal{L} ^{ij}(z) \mathcal{M} ^{jj}(z)- \mathcal{M} ^{ij}(z) \mathcal{L} ^{jj}(z)\bigr)_1={}& \operatorname{tr} _2\bigl(R_{12}^z(q_{ij}+\eta)P_{12}( \mathcal{S} ^{ij}J^\eta( \mathcal{S} ^{jj})- J^{\eta,q_{ij}}( \mathcal{S} ^{ij}) \mathcal{S} ^{jj})_2\bigr)-{} \\ &{}- \operatorname{tr} \mathcal{S} ^{jj}\cdot \operatorname{tr} _2\bigl(F_{12}^z(q_{ij}+\eta)P_{12} \mathcal{S} _2^{ij}\bigr). \end{aligned} $$

(55)

Collecting terms (48), (54), and (55), we obtain the $ij$th block of the commutator:

$$\begin{aligned} \, &\bigl([ \mathcal{L} (z), \mathcal{M} (z)]^{ij}\bigr)_1=( \operatorname{tr} \mathcal{S} ^{ii}- \operatorname{tr} \mathcal{S} ^{jj})\, \operatorname{tr} _2\bigl(F_{12}^z(q_{ij}+\eta)P_{12} \mathcal{S} _2^{ij}\bigr)+ \operatorname{tr} _2\bigl(R_{12}^z(q_{ij}+\eta)P_{12}A_2\bigr), \\ &A= \mathcal{S} ^{ii}J^{\eta,q_{ij}}( \mathcal{S} ^{ij})-J^\eta( \mathcal{S} ^{ii}) \mathcal{S} ^{ij}+ \mathcal{S} ^{ij}J^\eta( \mathcal{S} ^{jj})-J^{\eta,q_{ij}}( \mathcal{S} ^{ij}) \mathcal{S} ^{jj}+{} \\ &\phantom{A={}}+\sum_{k\colon k\ne i,j}\bigl( \mathcal{S} ^{ik}J^{\eta,q_{kj}} ( \mathcal{S} ^{kj})-J^{\eta,q_{ik}}( \mathcal{S} ^{ik}) \mathcal{S} ^{kj}\bigr). \end{aligned} $$

(56)

Also taking the last term (with $\mu^0_i$) in the right-hand side of (41) into account, we obtain the second equation in (20) in the form

$$\begin{aligned} \dot{ \mathcal{S} }^{ij}={}& \mathcal{S} ^{ii}J^{\eta,q_{ij}}( \mathcal{S} ^{ij})-J^\eta( \mathcal{S} ^{ii}) \mathcal{S} ^{ij}+ \mathcal{S} ^{ij}J^\eta( \mathcal{S} ^{jj})-J^{\eta, q_{ij}}( \mathcal{S} ^{ij}) \mathcal{S} ^{jj}+{} \\ &{}+\sum_{k\colon k\ne i,j}^M\bigl( \mathcal{S} ^{ik}J^{\eta,q_{kj}}( \mathcal{S} ^{kj})- J^{\eta,q_{ik}}( \mathcal{S} ^{ik}) \mathcal{S} ^{kj}\bigr). \end{aligned} $$

(57)

Here, we must clarify the transition from (56) to (57). Strictly speaking, we proved that the Lax equations hold on the equations of motion, but we did not prove the converse. To prove the converse, we must verify that all components of matrix equation (57) are independently contained in (56) taking into account that $R_{12}$ is a linear operator that somehow mixes these components in linear combinations. In other words, we must show that it follows from $\operatorname{tr} _2(R_{12}^z(q_{ij}+\eta) P_{12}C_2)=0$ that $C=0$. For this, we consider the Lax equation near $z=0$. It follows from (13)–(15) that $R_{12}^z(q_{ij}+\eta)P_{12}$ has a simple pole at $z=0$ with the residue equal to $P_{12}$. The needed statement then follows from the fact that $\operatorname{tr} _2(P_{12}A_2)=A$.

2.2.2. Diagonal part

We now consider equation (40), whose left-hand side has the form

$$\dot{ \mathcal{L} }_{ii}(z)_1= \operatorname{tr} _2\bigl(R_{12}^z(\eta)P_{12}\dot{ \mathcal{S} }^{ii}_2\bigr). $$

(58)

Using (34), we transform the expression in the summation in the right-hand side of (40):

$$\begin{aligned} \, \bigl( \mathcal{L} ^{ik}(z) \mathcal{M} ^{ki}(z)&{}- \mathcal{M} ^{ik}(z) \mathcal{L} ^{ki}(z)\bigr)_1= \nonumber \\ &\;= \operatorname{tr} _{23}\bigl[-R_{12}^z(q_{ik}+\eta)P_{12} \mathcal{S} _2^{ik}R_{13}^z(q_{ki}) P_{13} \mathcal{S} _3^{ki}+R_{12}^z(q_{ik})P_{12} \mathcal{S} _2^{ik} R_{13}^z(q_{ki}+\eta)P_{13} \mathcal{S} _3^{ki}\bigr]=\qquad \nonumber \\ &\;= \operatorname{tr} _{23}\bigl[(R_{12}^z(q_{ik})R_{23}(q_{ki}+\eta)- R_{12}^z(q_{ik}+\eta)R_{23}^z(q_{ki})) P_{12} \mathcal{S} _2^{ik}P_{13} \mathcal{S} _3^{ki}\bigr]\stackrel{(34)}{=} \nonumber \\ &\stackrel{(34)}{=} \operatorname{tr} _{23}\bigl[R_{13}^z(\eta)(r_{12}(q_{ik})- r_{12}(q_{ik}+\eta))P_{12} \mathcal{S} _2^{ik}P_{13} \mathcal{S} _3^{ki}\bigr]+{} \nonumber \\ &\hphantom{\;\;={}}+ \operatorname{tr} _{23}\bigl[(r_{23}(q_{ki}+\eta)-r_{23}(q_{ki})) R_{13}^z(\eta)P_{12} \mathcal{S} _2^{ik}P_{13} \mathcal{S} _3^{ki}\bigr]= \nonumber \\ &\;= \operatorname{tr} _2\bigl[R_{12}^z(\eta)P_{12}( \mathcal{S} ^{ik}J^{\eta,q_{ki}}( \mathcal{S} ^{ki})- J^{\eta,q_{ik}}( \mathcal{S} ^{ik}) \mathcal{S} ^{ki})_2\bigr]. \end{aligned}$$

(59)

Using (39), we simplify the rest of the right-hand side of (40):

$$\begin{aligned} \bigl( \mathcal{L} ^{ii}(z) \mathcal{M} ^{ii}(z)&{}- \mathcal{M} ^{ii}(z) \mathcal{L} ^{ii}(z)\bigr)_1= \\ &\;= \operatorname{tr} _{23}\bigl[-R_{12}^z(\eta)P_{12} \mathcal{S} _2^{ii}R_{13}^{(0),z} P_{13} \mathcal{S} _3^{ii}+R_{12}^{(0),z}P_{12} \mathcal{S} _2^{ii}R_{13}^z(\eta) P_{13} \mathcal{S} _3^{ii}\bigr]= \\ &\;= \operatorname{tr} _{23}\bigl[\bigl(R_{12}^{(0),z}R_{23}^z(\eta)- R_{12}^z(\eta)R_{23}^{(0),z}\bigr) P_{12} \mathcal{S} _2^{ii}P_{13} \mathcal{S} _3^{ii}\bigr]\stackrel{(39)}{=} \\ &\stackrel{(39)}{=} \operatorname{tr} _{23}\bigl[R_{13}^z(\eta) \bigl(r_{12}^{(0)}-r_{12}(\eta)\bigr) P_{12} \mathcal{S} _2^{ii}P_{13} \mathcal{S} _3^{ii}\bigr]+{} \\ &\hphantom{\;\;={}}+ \operatorname{tr} _{23}\bigl[\bigl(r_{23}(\eta)- r_{23}^{(0)}\bigr)R_{13}^z(\eta)P_{12} \mathcal{S} _2^{ii}P_{13} \mathcal{S} _3^{ii}\bigr]+{} \\ &\hphantom{\;\;={}}+ \operatorname{tr} _{23}\bigl[F_{13}^z(\eta)P_{12}P_{12} \mathcal{S} _2^{ii}P_{13} \mathcal{S} _3^{ii}\bigr]- \operatorname{tr} _{23}\bigl[P_{23}F_{13}^z(\eta) P_{12} \mathcal{S} _2^{ii}P_{13} \mathcal{S} _3^{ii}\bigr]. \end{aligned} $$

(60)

We note that the two last terms are equal and therefore cancel:

$$\operatorname{tr} _{23}\bigl(F_{13}^z(\eta)P_{12}P_{12} \mathcal{S} _2^{ii}P_{13} \mathcal{S} _3^{ii}\bigr)= \operatorname{tr} _{23}\bigl(P_{23}F_{13}^z(\eta)P_{12} \mathcal{S} _2^{ii}P_{13} \mathcal{S} _3^{ii}\bigr)= \operatorname{tr} \mathcal{S} ^{ii} \operatorname{tr} _2\bigl(F_{12}^z(\eta)P_{12} \mathcal{S} _2^{ii}\bigr). $$

(61)

The first and the second terms in (60) have the form

$$\begin{aligned} \, \operatorname{tr} _{23}\bigl[R_{13}^z(\eta)\bigl(r_{12}^{(0)}-r_{12}(\eta)\bigr) P_{12} \mathcal{S} _2^{ii}P_{13} \mathcal{S} _3^{ii}\bigr]+ \operatorname{tr} _{23} \bigl[\bigl(r_{23}(\eta)-r_{23}^{(0)}\bigr)R_{13}^z(\eta) P_{12} \mathcal{S} _2^{ii}P_{13} \mathcal{S} _3^{ii}\bigr]=& \nonumber \\ = \operatorname{tr} _2\bigl[R_{12}^z(\eta)P_{12} \bigl(J^\eta( \mathcal{S} ^{ii}) \mathcal{S} ^{ii}-{}& \mathcal{S} ^{ii}J^\eta( \mathcal{S} ^{ii})\bigr)_2\bigr]. \end{aligned}$$

(62)

From (59) and (62), we finally obtain

$$\bigl([L(z),M(z)]_{ii}\bigr)_1= \operatorname{tr} _2(R_{12}^z(\eta)P_{12}B_2), $$

(63)

where

$$B=J^\eta( \mathcal{S} ^{ii}) \mathcal{S} ^{ii}- \mathcal{S} ^{ii}J^\eta( \mathcal{S} ^{ii})+ \sum_{k\ne i}\bigl( \mathcal{S} ^{ik}J^{\eta,q_{ki}}( \mathcal{S} ^{ki})- J^{\eta, q_{ik}}( \mathcal{S} ^{ik}) \mathcal{S} ^{ki}\bigr). $$

(64)

Here, we should also use the argument given after (57). We have thus verified the equations of motion for the diagonal blocks.

2.3. Interacting tops

As explained in [3], in the particular case $\mathrm{rk}( \mathcal{S} )=1$, we can write the equations of motion in terms of only the diagonal blocks. We recall the main idea. The additional property $\mathrm{rk}( \mathcal{S} )=1$ yields

$$\mathcal{S} ^{ik}_1 P_{12} \mathcal{S} ^{ki}_1= \mathcal{S} ^{ii}_1 \mathcal{S} ^{kk}_2. $$

(65)

Further, for an arbitrary $J(S)= \operatorname{tr} _2(J_{12}S_2)$ of form (12) and ${\breve J}_{12}=J_{12}P_{12}$, we have

$$J(S)= \operatorname{tr} _2(J_{12}S_2)= \operatorname{tr} _2({\breve J}_{12}P_{12}S_2)= \operatorname{tr} _2(S_2{\breve J}_{12}P_{12})= \operatorname{tr} _2(S_2P_{12}{\breve J}_{21})= \operatorname{tr} _2(P_{12}S_1{\breve J}_{21}). $$

(66)

Therefore,

$$\mathcal{S} ^{ik}J( \mathcal{S} ^{ki})= \operatorname{tr} _2( \mathcal{S} ^{ik}_1P_{12} \mathcal{S} ^{ki}_1{\breve J}_{21})= \mathcal{S} ^{ii} \operatorname{tr} _2({\breve J}_{21} \mathcal{S} ^{kk}_2), $$

(67)

where ${\breve J}_{21}=P_{12}{\breve J}_{21}P_{12}=P_{12}J_{12}$. Similarly,

$$J( \mathcal{S} ^{ik}) \mathcal{S} ^{ki}= \operatorname{tr} _2({\breve J}_{12} \mathcal{S} ^{ik}_1P_{12} \mathcal{S} ^{ki}_1)= \operatorname{tr} _2({\breve J}_{12} \mathcal{S} ^{kk}_2) \mathcal{S} ^{ii}. $$

(68)

We finally write Eqs. (28) and (30) in the forms

$$\begin{aligned} \, &{\dot{ \mathcal{S} }}^{ii}=[ \mathcal{S} ^{ii},J^\eta( \mathcal{S} ^{ii})]+\sum_{k\colon k\ne i}^M \bigl( \mathcal{S} ^{ii}{\widetilde J}^{\eta,q_{ki}}(S^{kk})- {\breve J}^{\eta,q_{ik}}( \mathcal{S} ^{kk})S^{ii}\bigr), \\ &{\ddot q}_i= \operatorname{tr} ({\dot{ \mathcal{S} }}^{ii})=\sum_{k\colon k\ne i}^M \operatorname{tr} \bigl( \mathcal{S} ^{ii}{\widetilde J}^{\eta,q_{ki}}(S^{kk})- {\breve J}^{\eta,q_{ik}}( \mathcal{S} ^{kk})S^{ii}\bigr), \end{aligned} $$

(69)

where

$$\begin{aligned} \, &{\widetilde J}^{\eta,q_{ki}}( \mathcal{S} ^{kk})= \operatorname{tr} _2({\breve J}_{21}^{\eta,q_{ki}} \mathcal{S} ^{kk}_2 )= \operatorname{tr} _2(P_{12}J_{12}^{\eta,q_{ki}} \mathcal{S} ^{kk}_2), \\ &{\breve J}^{\eta,q_{ik}}( \mathcal{S} ^{kk})= \operatorname{tr} _2({\breve J}_{12}^{\eta,q_{ik}} \mathcal{S} ^{kk}_2 )= \operatorname{tr} _2({J}_{12}^{\eta,q_{ik}}P_{12} \mathcal{S} ^{kk}_2). \end{aligned} $$

(70)

Written in form (69), the equations of motion are interpreted as the dynamical equations for $M$ particles with additional “spin” degrees of freedom, i.e., the particles can be identified with tops that also have coordinates and velocities in addition to their own internal degrees of freedom. The interaction between the tops depends on both the distance and the spin dynamical variables.