  We consider $(n+1)$ bodies moving under their mutual gravitational attraction
in spaces with constant Gaussian curvature $\kappa$. In this system, $n$
primary bodies with equal masses form a relative equilibrium solution with a
regular polygon configuration, and the remaining body of negligible mass does
not affect the motion of the others. We show that the singularity due to binary
collision between the negligible mass and the primaries can be regularized
local and globally through suitable changes of coordinates (Levi-Civita and
Birkhoff type transformations).
