I came across a physical system which obeys the following ODE $$\frac{d z_n}{dt} = \sum_{m=1}^N i M_{nm} \frac{z_m - z_n}{|z_m - z_n|}, \qquad n\in\{1,2,\dots,N\}$$ where $z_n \equiv z_n(t)$ are complex valued functions of time and $M_{nm}$ is a symmetric positive definite real matrix. The system is clearly nonlinear so I suspect it doesn't have an exact solution.
Is there a fixed point? Is there a way to understand the evolution in the close neighbourhood of $z_n = 0$? Note that the solution is nontrivial only if the matrix is not diagonal!