The meromorphic solutions of the Bruschi-Calogero equation (Q1577251)

This paper is devoted to the functional equations \[ \alpha(x) \alpha'(y)- \alpha'(x) \alpha(y)= \bigl(\alpha (x+y)-\alpha (x)\alpha(y) \bigr) \bigl(\eta(x) -\eta(y) \bigr), \] which is called the Bruchi-Calogero equation, where the function \(v(x)\) is given by \[ v(x)= {d\over dx}\log \bigl(\alpha (x) \alpha(-x) -1\bigr). \] Note that the Bruchi-Calogero equation is obtained from the Lax representation \(\dot L=[L,M])\) of the equation of motion \[ \ddot q_j= \sum^n_{k=1 \atop k\neq j} \dot q_j\dot q_k v(q_j-q_k),\quad q_j=q_j(t),\;j=1, \dots,n \] which is called Ruijsenaars and Schneider type equation. The authors give all meromorphic solutions of the Bruschi-Calogero equation (1) defined near the origin \(0\in\mathbb{C}\).