  The goal of this paper is to give a detailed analytical description of the
global dynamics of N points interacting through the singular logarithmic
potential and subject to the following symmetry constraint: at each instant
they form an orbit of the dihedral group of order 2l. The main device in order
to achieve our results is a technique very popular in Celestial Mechanics,
usually referred to as "McGehee transformation". After performing this change
of coordinates that regularizes the total collision, we study the rest-points
of the flow, the invariant manifolds and we derive interesting information
about the global dynamics for l=2. We observe that our problem is equivalent to
studying the geometry of stationary configurations of nearly-parallel vortex
filaments in three dimensions in the LIA approximation.
