  We construct a non-perturbative, single-valued solution for the metric and
the motion of $N$ interacting particles in $2+1$-Gravity. The solution is
explicit for two particles with any speed and for any number of particles with
small speed. It is based on a mapping from multivalued Minkowskian coordinates
to single-valued ones, which solves the non-abelian monodromies due to
particles' momenta. The two and three-body cases are treated in detail.
