Studying the dynamics of the endpoints of an equilibrium measure (a minimizer of its logarithmic energy in an external field) I ran into the following system of differential equations (which I state for the case of 4 points, for simplicity): let $x_j=x_j(t)$, $j=1, \dots, 4$, be real values dependent on time $t$, all distinct at $t=0$, and satisfying the system $$ \frac{d x_j}{dt} = \frac{m_j}{q'(x_j)}=m_j \prod_{k\neq j} (x_j-x_k)^{-1}, \quad j=1, \dots, 4, $$ where $q(x)=\prod_{j=1}^4 (x-x_j)$ and $q'(x)$ is its derivative with respect to $x$. Here $m_j$ are positive numbers.

My questions (sorry if too elementary or naive) are:

1) is this kind of a system known, does it have any name attached to it?

2) I needed to prove the fact that the interior $x_j$'s collide in a finite time. Does this follow from any general fact in dynamical systems or systems of ODEs?

3) what about the more general situation, when the number of points is $n$ and the right hand sides in the system are rational functions?

Thanks in advance.