Classical elliptic ${\rm BC}_1$ Ruijsenaars–van Diejen model: relation to Zhukovsky–Volterra gyrostat and 1-site classical $XYZ$ model with boundaries

A. M. Mostowskii, A. V. Zotov

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

Abstract: We present a description of the classical elliptic $\mathrm{BC}_1$ Ruijsenaars–van Diejen model with eight independent coupling constants through a pair of $\mathrm{BC}_1$ type classical Sklyanin algebras generated by the (classical) quadratic reflection equation with non-dynamical $XYZ$ $r$-matrix. For this purpose, we consider the classical version of the $L$-operator for the Ruijsenaars–van Diejen model proposed by O. Chalykh. In the $\mathrm{BC}_1$ case, it is factorized into the product of two Lax matrices depending on four constants. Then we apply an IRF-Vertex type gauge transformation and obtain a product of the Lax matrices for the Zhukovsky–Volterra gyrostats. Each of them is described by the $\mathrm{BC}_1$ version of the classical Sklyanin algebra. In particular case, when four pairs of constants coincide, the $\mathrm{BC}_1$ Ruijsenaars–van Diejen model exactly coincides with the relativistic Zhukovsky–Volterra gyrostat. Explicit change of variables is obtained. We also consider another special case of the $\mathrm{BC}_1$ Ruijsenaars–van Diejen model with seven independent constants. We show that it can be reproduced by considering the transfer matrix of the classical $1$-site $XYZ$ chain with boundaries. In the end of the paper, using another gauge transformation, we represent Chalykh's Lax matrix in a form depending on Sklyanin's generators.

Keywords: integrable systems, quadratic Poisson algebras, Zhukovsky–Volterra gyrostat.

Received: 27.10.2025
Revised: 27.10.2025

DOI: 10.4213/tmf11101

 English version: Theoretical and Mathematical Physics, 2026, 226:2, 189–216