On the integrability of classical Ruijsenaars-Schneider Model of BC2 type

The problem of finding most general form of the classical integrable relativistic models of many-body interaction of the BCn type is considered. In the simplest nontrivial case of n=2, the extra integral of motion is presented in explicit form within the ansatz similar to the nonrelativistic Calogero-Moser models. The resulting Hamiltonian has been found by solving the set of two functional equations. 1 permanent address: BLTP JINR, 141980 Dubna, Moscow Region, Russia The one-parameter extensions of the Calogero-Moser integrable particle systems first proposed by Ruijsenaars and Schneider [1,2] have been extensively studied during last decade [3,4,6-17]. It turned out that both classical and quantum versions of these systems (often called “relativistic” but, as explained in [6], this term is misleading) were connected with various branches of mathematical physics. The quantum RS models with trigonometric type of interaction have been found to be relevant in the theory of Macdonald polynomials and the integrability of their elliptic version has been investigated within the Yang-Baxter approach [8,9]. The applications of classical models range from characteristics of multisoliton solutions of the sine-Gordon equation to the Seiberg-Witten theory [13]. The original n-particle RS model has the Hamiltonian Hn = n