A family of integrable and non-integrable difference equations arising from cluster algebras

Atsushi Nobe, Junta Matsukidaira

The one-parameter family of second order nonlinear difference equations each of which is given by $$x_{n-1}{x}_n{x}_{n+1}=x_{n-1}+(x_n{)}^{\beta -1}+x_{n+1}(\beta \in \mathbb{N})$$ is expl…