Abstract
We present a description of the classical elliptic \({\rm BC}_1\) Ruijsenaars–van Diejen model with eight independent coupling constants through a pair of \({\rm BC}_1\) 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 \({\rm BC}_1\) 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 \({\rm BC}_1\) version of the classical Sklyanin algebra. In particular case, when four pairs of constants coincide, the \({\rm BC}_1\) 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 \({\rm BC}_1\) 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.
