I am interested in solving the following system of differential equations \begin{equation} \ddot{z}_j = \sum_{k\neq j}^n\frac{2\dot{z}_j\dot{z}_k}{z_j-z_k},\qquad \forall j\in\{1,\dots,n\}, \end{equation} for $n$ even and initial conditions satisfying \begin{equation} {z}_{j+n/2}(0) = {z}_{j}^{-1}(0),\qquad \forall j\in\{1,\dots,n/2\}. \end{equation}
I have found the following solution that works except for the fact that it does not satisfy the initial conditions (for $n>2$): \begin{equation} {z}_{j}(t) = a + b e^{2\pi i \frac{j}{n}}(t-c)^{\frac{1}{n}}. \end{equation}
I am looking for a way to modify this solution in order to be able to satisfy the initial condition, but haven't been successful. Any hints or references would be very welcome.