  We construct a non-perturbative, single-valued solution for the metric and
the motion of two interacting particles in ($2+1$)-Gravity, by using a Coulomb
gauge of conformal type. The method provides the mapping from multivalued (
minkowskian ) coordinates to single-valued ones, which solves the non-abelian
monodromies due to particles's momenta and can be applied also to the general
N-body case.
