the values of $h_{n+1}, \dots, h_{2n}$ at any $A \in U$ are determined by $h_1(A), \dots, h_n(A)$, and similarly for $\tilde{B} \in \eta U \eta^{-1}$. Thus, we have proved that the intersection $\mathcal{C}_{**} \cap W$ is the joint zero set of the functions $F_i \in \mathcal{C}^{\infty}(W, \mathbb{R})$ defined by

$$F_i(A, \tilde{B}, v_1, \dots, v_d) := h_i(A) - h_i(\tilde{B}), \quad i = 1, \dots, n, \qquad (B.11)$$

whose differentials are linearly independent on $W$. By a standard result of differential geometry [55], this implies that the subset $\mathcal{C}_{**}$ inherits a unique manifold structure from $M_{d**}$, such that it also has the subspace topology. $\square$

### C. Poisson Algebra of Invariant First Integrals and a Conjecture

Consider $M_d$ equipped with the quasi-Poisson bracket associated with the quasi-Poisson bivector $P_{\bar{z}}$ defined in (3.19) for a fixed $\bar{z} = (z_{\alpha\beta})_{\alpha<\beta} \in \mathbb{R}^{d(d-1)/2}$, for any $d \ge 2$. Recall from Sect. 4.1 that $U(n)$-invariant functions of the matrix $A$ are Poisson commuting on $M_d$, and together with their first integrals (constants of motion) they satisfy a property analogous to being degenerately integrable. Motivated by the original considerations of Gibbons-Hermsen [23] and the investigation of Marshall and the authors in the Poisson-Lie setting [14], we are led to consider among all the $U(n)$-invariant first integrals those of a peculiar form: they are obtained by taking traces of matrices involving solely $A$ and the spin variables $v_1, \dots, v_d$. The Poisson algebra $\mathfrak{I}$ (C.2) generated by such first integrals contains in its Poisson center the algebra $\mathfrak{H}_{\text{tr}}$ of polynomials in the functions $\Re(\text{tr } A^k), \Im(\text{tr } A^k), k \in \mathbb{Z}$; in analogy with [14] we conjecture that the pair $(\mathfrak{I}, \mathfrak{H}_{\text{tr}})$ yields a degenerate integrable system on the reduced phase space $\Phi^{-1}(e^{i\gamma})/U(n)$ considered in Sect. 6 (for generic $\gamma \in \mathbb{R}$), though we can not prove this result at present.

We start by working in the Poisson algebra $\mathcal{C}^{\infty}(M_d)^{U(n)}$ obtained by restricting the quasi-Poisson bracket (associated with the quasi-Poisson bivector $P_{\bar{z}}$ from (3.19)) on $\mathcal{C}^{\infty}(M_d)$ to invariant functions. We let $\mathfrak{H}_{\text{tr}} := \mathbb{R}[\Re(\text{tr } A^k), \Im(\text{tr } A^k) | k \in \mathbb{Z}]$, and we define

$$\mathfrak{J}_0 := \{F(x_1|v_1|^2, \dots, x_d|v_d|^2) \mid F(t_1, \dots, t_d) \text{ is real analytic on } (-2\pi, 2\pi)^{\times d}\}. \qquad (C.1)$$

In particular, $e^{|v_\alpha|^2}$ or $\mathfrak{b}(x_\alpha|v_\alpha|^2)$ with $\mathfrak{b}(t)$ defined in (2.23) belong to $\mathfrak{J}_0$ for any $1 \le \alpha \le d$. We put $I_{\alpha\beta}^k := v_\alpha^\dagger A^k v_\beta (=\text{tr}(v_\beta v_\alpha^\dagger A^k))$ and we introduce the polynomial algebra generated by the real and imaginary parts of the $I_{\alpha\beta}^k$ and $\text{tr}(A^k)$ with coefficients in $\mathfrak{J}_0$:

$$\mathfrak{J} := \mathfrak{J}_0[\Re(\text{tr } A^k), \Im(\text{tr } A^k), \Re(I_{\alpha\beta}^k), \Im(I_{\alpha\beta}^k) | k \in \mathbb{Z}_{\ge 0}, 1 \le \alpha, \beta \le d]. \quad (C.2)$$

Note that $I_{\alpha\alpha}^0 = |v_\alpha|^2 \in \mathfrak{J}_0 \subset \mathfrak{J}$ for all $1 \le \alpha \le d$ and $\mathfrak{H}_{\text{tr}} \subset \mathfrak{J}$. When defining $\mathfrak{J}$, we could allow any $k \in \mathbb{Z}$ since $A$ is unitary and $I_{\alpha\beta}^{-k} = \overline{I_{\beta\alpha}^k}$. We also remark that, while $\mathfrak{J}$ is infinite-dimensional, it is finitely generated over $\mathfrak{J}_0$: we only
need to consider functions tr $A^k$, $I_{\alpha\beta}^k$ with exponent $k \in \{1, \dots, n\}$ due to the Cayley-Hamilton theorem for $A$.

**Proposition C.1.** For the parameters $\underline{z} = (z_{\alpha\beta})_{\alpha<\beta}$ defining the quasi-Poisson bivector $P_{\underline{z}}$ as in (3.19), consider their antisymmetric extension $(z_{\alpha\beta}^*)_{1\le\alpha,\beta\le d}$ obtained by setting $z_{\alpha\alpha}^* = 0$ and $z_{\alpha\beta}^* = -z_{\beta\alpha}^* := z_{\alpha\beta}$ for $\alpha < \beta$. Denote by $\{-, -\}_{\underline{z}}$ the quasi-Poisson bracket associated with $P_{\underline{z}}$.

(1) $\{\text{tr}(A^k), \text{tr}(A^l)\}_{\underline{z}} = 0$ and $\{I_{\alpha\beta}^k, \text{tr}(A^l)\}_{\underline{z}} = 0$ for any $k, l \in \mathbb{Z}_{\ge 0}$ and $1 \le \alpha, \beta \le d$.

(2) For any $k, l \in \mathbb{Z}_{\ge 0}$ and $1 \le \alpha, \beta, \gamma, \epsilon \le d$, the following holds

$$ \begin{aligned} \{I_{\alpha\beta}^k, I_{\gamma\epsilon}^l\}_{\underline{z}} ={}& \frac{1}{2} I_{\gamma\beta}^0 I_{\alpha\epsilon}^{k+l} - \frac{1}{2} I_{\gamma\beta}^{k+l} I_{\alpha\epsilon}^0 + \frac{1}{2} \left( \sum_{1 \le r \le k} - \sum_{1 \le r \le l} \right) (I_{\gamma\beta}^r I_{\alpha\epsilon}^{k+l-r} - I_{\gamma\beta}^{k+l-r} I_{\alpha\epsilon}^r) \\ & + \frac{1}{2} \operatorname{sgn}(\gamma - \alpha) I_{\gamma\beta}^k I_{\alpha\epsilon}^l - z_{\alpha\gamma}^* I_{\alpha\beta}^k I_{\gamma\epsilon}^l + \frac{1}{2} \operatorname{sgn}(\epsilon - \beta) I_{\gamma\beta}^l I_{\alpha\epsilon}^k - z_{\beta\epsilon}^* I_{\alpha\beta}^k I_{\gamma\epsilon}^l \\ & - \frac{1}{2} \operatorname{sgn}(\epsilon - \alpha) I_{\gamma\beta}^{k+l} I_{\alpha\epsilon}^0 + z_{\alpha\epsilon}^* I_{\alpha\beta}^k I_{\gamma\epsilon}^l - \frac{1}{2} \operatorname{sgn}(\gamma - \beta) I_{\gamma\beta}^0 I_{\alpha\epsilon}^{k+l} + z_{\beta\gamma}^* I_{\alpha\beta}^k I_{\gamma\epsilon}^l \\ & + \frac{i}{x_\beta} \delta_{\beta\gamma} I_{\gamma\epsilon}^{k+l} + \frac{i}{2} \delta_{\beta\gamma} b(x_\beta |v_\beta|^2) [ |v_\beta|^2 I_{\alpha\epsilon}^{k+l} - I_{\alpha\beta}^k I_{\gamma\epsilon}^l ] \\ & - \frac{i}{x_\alpha} \delta_{\alpha\epsilon} I_{\gamma\beta}^{k+l} - \frac{i}{2} \delta_{\alpha\epsilon} b(x_\alpha |v_\alpha|^2) [ |v_\alpha|^2 I_{\gamma\beta}^{k+l} - I_{\alpha\beta}^k I_{\gamma\epsilon}^l ], \end{aligned} \quad (C.3) $$

where we omit the first line if $k=0$ or $l=0$.

(3) $\{|v_\alpha|^2, I_{\gamma\epsilon}^l\}_{\underline{z}} = \frac{i}{x_\alpha}(\delta_{\alpha\gamma} - \delta_{\alpha\epsilon})I_{\gamma\epsilon}^l$, hence $\{|v_\alpha|^2, |v_\gamma|^2\}_{\underline{z}} = 0$ for any $l \in \mathbb{Z}_{\ge 0}$, $1 \le \alpha, \gamma, \epsilon \le d$.

In particular, the subalgebra $\mathfrak{I} \subset \mathcal{C}^\infty(M_d)^{U(n)}$ is a Poisson algebra whose center contains $\mathfrak{H}_{\text{tr}}$.

*Proof.* We have (1) directly from Sect. 4.1. We deduce (3) from (2) by taking $k=0$ then $l=0$ and $\gamma = \epsilon$. In view of these three items, $\mathfrak{I} \subset \mathcal{C}^\infty(M_d)^{U(n)}$ is a Poisson algebra. Moreover, the Poisson center of $\mathfrak{I}$ contains $\mathfrak{H}_{\text{tr}}$ by (1).

To get (2), recall that we have the decomposition $\{-, -\}_{\underline{z}} = \{-, -\} + \{-, -\}_{\underline{\psi}_z}$, where $\{-, -\}$ is the bracket explicitly presented in Theorem 3.1, while $\{-, -\}_{\underline{\psi}_z}$ satisfies (3.20) and (3.21). Deriving (C.3) is therefore a standard computation after decomposing the elements $I_{\alpha\beta}^k = \sum_{1 \le i,j \le n} (\bar{v}_\alpha)_i (A^k)_{ij} (v_\beta)_j$ in terms of entries of the matrices defining $M_d$. $\square$

When $x_1 = \dots = x_d = 1$ and $z_{\alpha\beta}^* = \operatorname{sgn}(\alpha - \beta)$ for any $1 \le \alpha, \beta \le d$, Proposition C.1 yields a real version of the algebra of first integrals given by Chalykh and the first-named author in [10, §5.1]. This choice of parameters $(z_{\alpha\beta}^*)$ corresponds to performing successive fusions of the residual U(1)-actions as we explained in Sect. 3.2.3, which correspond to analogous fusions of $\mathcal{C}^\times$-actions performed in [10]. In fact, these Poisson algebras can be compared through the formulas (C.3) and [10, (5.1)]: they match up to an overall sign (due to a different choice of trace pairing) except for the terms in $\delta_{\alpha\epsilon}$ and $\delta_{\beta\gamma}$ whose discrepancy is explained by the (local) change of variables given in Remark A.8.
On the other hand, we recall that in the complex case, besides the one
in [10], there is another Poisson algebra that was derived by Arutyunov and
Olivucci [6] from a Poisson–Lie perspective. While both algebras describe the
first integrals for the trigonometric/hyperbolic spin RS system in the complex
setting, it is a puzzling observation that they do not match. Nevertheless, if we
take our formula (C.3) in the case where $x_1 = \dots = x_d = 1$ and all $z_{\alpha\beta} = 0$,
then comparison with [6, Eq. (7.5)] (choosing $\kappa = 1$ and the ‘minus’ Poisson
bracket) shows that the two Poisson algebras have the same form.<fnref n="8" /> Building
on these observations, we expect that the Poisson algebras of [6] and [10] can
be simultaneously considered in the following way.

**Conjecture C.2.** There exists a pencil of compatible Poisson brackets on the phase space of the complex trigonometric spin RS system (denoted $M_{n,d,q}^\times$// $GL_n(\mathbb{C})$ in [10]) which contains the Poisson structures of Arutyunov–Olivucci [6] and of Chalykh–Fairon [10] as special instances. Furthermore, for all such Poisson brackets, one can obtain a Hamiltonian formulation of the trigonometric spin RS system of Krichever and Zabrodin [30].

We are confident that the results presented in Sect. 3.2 can be adapted
to the complex setting to yield a pencil of compatible quasi-Poisson brackets
on $M_{n,d,q}^\times$, a special case of which is given in [10]. We also believe that Conjec-
ture C.2 can be approached from a Poisson–Lie perspective by endowing the
oscillator manifold of [6] with a pencil of compatible Poisson brackets.

References

[1] Alekseev, A., Kosmann-Schwarzbach, Y., Meinrenken, E.: Quasi-Poisson manifolds. Canad. J. Math. **54**, 3–29 (2002). arXiv:math/0006168

[2] Alekseev, A., Malkin, A., Meinrenken, E.: Lie group valued moment maps. J. Differ. Geom. **48**, 445–495 (1998). arXiv:dg-ga/9707021

[3] Arthamonov, S., Reshetikhin, N.: Superintegrable systems on moduli spaces of flat connections. Commun. Math. Phys. **386**, 1337–1381 (2021). arXiv:1909.08682

[4] Arutyunov, G., Frolov, S.: On the Hamiltonian structure of the spin Ruijsenaars-Schneider model. J. Phys. A **31**, 4203–4216 (1998). arXiv:hep-th/9703119

[5] Arutyunov, G.E., Frolov, S.A., Medvedev, P.B.: Elliptic Ruijsenaars–Schneider model from the cotangent bundle over the two-dimensional current group. J. Math. Phys. **38** Paper No. 5682; (1997) arXiv:hep-th/9608013

[6] Arutyunov, G., Olivucci, E.: Hyperbolic spin Ruijsenaars-Schneider model from Poisson reduction. Proc. Steklov Inst. Math. **309**, 31–45 (2020). arXiv:1906.02619

[7] Bredon, G.E.: Introduction to Compact Transformation Groups. Academic Press, London (1972)

[8] Calogero, F.: Solution of the one-dimensional N-body problem with quadratic and/or inversely quadratic pair potentials. J. Math. Phys. **12**, 419–436 (1971)

<footnote n="8">⁸The comparison with [6] is rather subtle: one not only needs to complexify and make use of Remark A.8, but also needs to use an alternative choice of first integrals after complexification. See also [10, Remark 5.2].</footnote>
[9] Chalykh, O., Fairon, M.: Multiplicative quiver varieties and generalised Ruijsenaars-Schneider models. J. Geom. Phys. **121**, 413–437 (2017). arXiv:1704.05814

[10] Chalykh, O., Fairon, M.: On the Hamiltonian formulation of the trigonometric spin Ruijsenaars-Schneider system. Lett. Math. Phys. **110**, 2893–2940 (2020). arXiv:1811.08727

[11] Duistermaat, J.J., Kolk, J.A.C.: Lie Groups. Universitext. Springer, Berlin (2000)

[12] Etingof, P., Varchenko, A.: Geometry and classification of solutions of the classical dynamical Yang-Baxter equation. Commun. Math. Phys. **192**, 77–120 (1998). arXiv:q-alg/9703040

[13] Fairon, M.: Integrable systems on multiplicative quiver varieties from cyclic quivers. Preprint, arXiv:2108.02496

[14] Fairon, M., Fehér, L., Marshall, I.: Trigonometric real form of the spin RS model of Krichever and Zabrodin. Ann. Henri Poincaré **22**, 615–675 (2021). arXiv:2007.08388

[15] Fairon, M., Görbe, T.: Superintegrability of Calogero-Moser systems associated with the cyclic quiver. Nonlinearity **34**, 7662–7682 (2021). arXiv:2101.05520

[16] Fehér, L.: Poisson–Lie analogues of spin Sutherland models. Nucl. Phys. B **949**, Paper No. 114807, (2019) arXiv:1809.01529

[17] Fehér, L.: Bi-Hamiltonian structure of Sutherland models coupled to two $u(n)^*$-valued spins from Poisson reduction. Nonlinearity **35**, 2971–3003 (2022). arXiv:2109.07391

[18] Fehér, L.: Poisson reductions of master integrable systems on doubles of compact Lie groups. Ann. Henri Poincaré **24**, 1823–1876 (2023). arXiv:2208.03728

[19] Fehér, L., Klimčík, C.: Poisson–Lie interpretation of trigonometric Ruijsenaars duality. Commun. Math. Phys. **301**, 55–104 (2011). arXiv:0906.4198

[20] Fehér, L., Klimčík, C.: Self-duality of the compactified Ruijsenaars-Schneider system from quasi-Hamiltonian reduction. Nucl. Phys. B **860**, 464–515 (2012). arXiv:1101.1759

[21] Fehér, L., Kluck, T.J.: New compact forms of the trigonometric Ruijsenaars-Schneider system. Nucl. Phys. B **882**, 97–127 (2014). arXiv:1312.0400

[22] Fock, V., Gorsky, A., Nekrasov, N., Rubtsov, V.: Duality in integrable systems and gauge theories. JHEP **07**, Paper No. 028 (2000) arXiv:hep-th/9906235

[23] Gibbons, J., Hermsen, T.: A generalisation of the Calogero-Moser system. Physica D **11**, 337–348 (1984)

[24] Gotô, M.: A theorem on compact semi-simple groups. J. Math. Soc. Japan **1**, 270–272 (1949)

[25] Hurtubise, J., Jeffrey, L., Sjamaar, R.: Group-valued implosion and parabolic structures. Am. J. Math. **128**, 167–214 (2006). arXiv:math/0402464

[26] Jovanovic, B.: Symmetries and integrability. Publ. Inst. Math. **49**, 1–36 (2008). arXiv:0812.4398

[27] Kazhdan, D., Kostant, B., Sternberg, S.: Hamiltonian group actions and dynamical systems of Calogero type. Commun. Pure Appl. Math. **31**, 481–507 (1978)
[28] Kharchev, S., Levin, A., Olshanetsky, M., Zotov, A.: Quasi-compact Higgs bundles and Calogero-Sutherland systems with two types spins. J. Math. Phys. **59**, 103509 (2018)

[29] Kowalczyk-Murynka, K., Kuś, M.: Matrix and vectorial generalized Calogero-Moser models. Physica D **440**, 133491 (2022)

[30] Krichever, I., Zabrodin, A.: Spin generalization of the Ruijsenaars-Schneider model, non-abelian 2D Toda chain and representations of Sklyanin algebra. Russian Math. Surv. **50**, 1101–1150 (1995). arXiv:hep-th/9505039

[31] Laurent-Gengoux, C., Miranda, E., Vanhaecke, P.: Action-angle coordinates for integrable systems on Poisson manifolds. Int. Math. Res. Not. IMRN **2011**, 1839–1869 (2011)

[32] Le Blanc, A.: Quasi-Poisson structures and integrable systems related to the moduli space of flat connections on a punctured Riemann sphere. J. Geom. Phys. **57**, 1631–1652 (2007)

[33] Li, L.-C., Xu, P.: A class of integrable spin Calogero-Moser systems. Commun. Math. Phys. **231**, 257–286 (2002). arXiv:math/0105162

[34] Mischenko, A.S., Fomenko, A.T.: Generalized Liouville method for integrating Hamiltonian systems. Funct. Anal. Appl. **12**, 113–125 (1978)

[35] Moser, J.: Three integrable Hamiltonian systems connected with isospectral deformations. Adv. Math. **16**, 197–220 (1975)

[36] Nekhoroshev, N.N.: Action-angle variables and their generalizations. Trans. Moscow Math. Soc. **26**, 180–197 (1972)

[37] Nekrasov, N.: Infinite-dimensional algebras, many-body systems and gauge theories. Moscow Seminar in Mathematical Physics, AMS Transl. Ser. 2, Vol. 191, Amer. Math. Soc., Providence, RI, pp. 263–299, (1999)

[38] Olshanetsky, M.A., Perelomov, A.M.: Classical integrable finite-dimensional systems related to Lie algebras. Phys. Rept. **71**, 313–400 (1981)

[39] Ortega, J.-P., Ratiu, T.: Momentum Maps and Hamiltonian Reduction. Birkhäuser, (2004)

[40] Procesi, C.: Lie Groups. An Approach through Invariants and Representations. Universitext. Springer, Berlin (2007)

[41] Pusztai, B.G.: The hyperbolic $BC(n)$ Sutherland and the rational $BC(n)$ Ruijsenaars-Schneider-van Diejen models: Lax matrices and duality. Nucl. Phys. B **856**, 528–551 (2012). arXiv:1109.0446

[42] Reshetikhin, N.: Degenerate integrability of spin Calogero-Moser systems and the duality with the spin Ruijsenaars systems. Lett. Math. Phys. **63**, 55–71 (2003). arXiv:math/0202245

[43] Reshetikhin, N.: Degenerately integrable systems. J. Math. Sci. **213**, 769–785 (2016). arXiv:1509.00730

[44] Reshetikhin, N.: Spin Calogero-Moser models on symmetric spaces. Integrability, Quantization, and Geometry. I. Integrable Systems. In: Proceedings of Symposium Pure Math., 103.1, Amer. Math. Soc., Providence, RI, pp. 377–402, (2021) arXiv:1903.03685

[45] Reshetikhin, N., Schrader, G.: Superintegrability of generalized Toda models on symmetric spaces. Int. Math. Res. Not. IMRN **2021**, 12993–13010 (2011)

[46] Rudolph, G., Schmidt, M.: Differential Geometry and Mathematical Physics. Part I. Manifolds, Lie Groups and Hamiltonian Systems. Springer, Berlin (2013)
[47] Ruijsenaars, S.N.M.: Action-angle maps and scattering theory for some finite-dimensional integrable systems III. Sutherland type systems and their duals. Publ. RIMS 31, 247-353 (1995)

[48] Ruijsenaars, S.N.M., Schneider, H.: A new class of integrable systems and its relation to solitons. Ann. Phys. (NY) 170, 370-405 (1986)

[49] Sjamaar, R., Lerman, E.: Stratified symplectic spaces and reduction. Ann. Math. 134, 375-422 (1991)

[50] Śniatycki, J.: Differential Geometry of Singular Spaces and Reduction of Symmetries. Cambridge University Press, Cambridge (2013)

[51] Semenov-Tian-Shansky, M.A.: Integrable systems: an r-matrix approach. Kyoto preprint RIMS-1650, (2008); http://www.kurims.kyoto-u.ac.jp/preprint/file/RIMS1650.pdf

[52] Sutherland, B.: Exact results for a quantum many-body problem in one dimension. Phys. Rev. A 4, 2019-2021 (1971)

[53] Trunina, E., Zotov, A.: Lax equations for relativistic GL($NM$,C) Gaudin models on elliptic curve. J. Phys. A 55, Paper No. 395202, pp. 38 (2022) [arXiv:2204.06137]

[54] Van den Bergh, M.: Double Poisson algebras. Trans. Am. Math. Soc. 360, 5711-5769 (2008). [arXiv:math/0410528]

[55] Warner, F.W.: Foundations of Differentiable Manifolds and Lie Groups. Springer, Berlin (1983)

[56] Wojciechowski, S.: An integrable marriage of the Euler equations with the Calogero-Moser system. Phys. Lett. A 111, 101-103 (1985)

M. Fairon

Laboratoire de Mathématiques d'Orsay

Université Paris-Saclay

91405 Orsay

France

e-mail: maxime.fairon@universite-paris-saclay.fr

L. Fehér

Department of Theoretical Physics

University of Szeged

Tisza Lajos krt 84-86

6720 Szeged

Hungary

and

Wigner Research Centre for Physics, Institute for Particle and Nuclear Physics

P.O.B. 49 1525 Budapest

Hungary

e-mail: lfeher@physx.u-szeged.hu

Communicated by Nikolai Kitanine.

Received: March 4, 2023.

Accepted: June 19, 2023.