A new goldfish model

A new integrable (indeed, solvable) model of goldfish type is identified, and some of its properties are discussed. A version of its Newtonian equations of motion reads as follows:
$$ \ddot{z}_n=\frac{\beta\dot{z}_n(\dot{z}_n+\eta)}{1+\beta z_n}+ \sum_{m=1,m\ne n}^N\frac{(\dot{z}_n-\beta\eta z_n) (\dot{z}_m-\beta\eta z_m)\bigl[2+\beta(z_n+z_m)\bigr]} {(z_n-z_m)(1+\beta z_m)}, $$
where $z_n\equiv z_n(t)$ are the $N$ dependent variables, $t$ is the independent variable (“time”), the dots indicate time differentiations, and $\beta$, $\eta$ are two arbitrary constants.