Another new goldfish model

A new integrable (indeed, solvable) model of goldfish type is identified, and some of its properties are discussed. Its Newtonian equations of motion read as follows:
\begin{align*} \ddot z_n={}&\frac{\dot z_n^2}{z_n}+c_1\frac{\dot z_n}{z_n}+ c_2\dot z_n+c_2c_3z_n+c_1c_2+{} \\[2mm] &{}+\sum_{m=1,m\ne n}^N\frac{(\dot z_n+c_3z_n+c_1)(\dot z_m+c_3z_m+c_1)} {z_m}\cdot\frac{z_n+z_m}{z_n-z_m},\quad n=1,\dots,N, \end{align*}
where $c_1$, $c_2$, and $c_3$ are arbitrary constants, $z_n\equiv z_n(t)$ are the $N$ dependent variables, $N$ is an arbitrary positive number $(N>1)$, $t$ is the independent variable {(}“time”{\rm)} and the dots indicate time-differentiations.