  In previous work, black hole vortex solutions in Einstein gravity with
AdS$_3$ background were found where the scalar matter profile had a singularity
at the origin $r=0$. In this paper, we find numerically static vortex solutions
where the scalar and gauge fields have a non-singular profile under Einstein
gravity in an AdS$_3$ background. Vortices with different winding numbers $n$,
VEV $v$ and cosmological constant $\Lambda$ are obtained. These vortices have
positive mass and are not BTZ black holes as they have no event horizon. The
mass is determined in two ways: by subtracting the numerical values of two
separate asymptotic metrics and via an integral that is purely over the matter
fields. The mass of the vortex increases as the cosmological constant becomes
more negative and this coincides with the core of the vortex becoming smaller
(compressed). We then consider the vortex with gravity in asymptotically flat
spacetime for different values of the coupling $\alpha=1/(16 \pi G)$. At the
origin, the spacetime has its highest curvature and there is no singularity. It
transitions to an asymptotic conical spacetime with angular deficit that
increases significantly as $\alpha$ decreases. For comparison, we also consider
the vortex without gravity in flat spacetime. For this case, one cannot obtain
the mass by the first method (subtracting two metrics) but remarkably, via a
limiting procedure, one can obtain an integral mass formula. In the absence of
gauge fields, there is a well-known logarithmic divergence in the energy of the
vortex. With gravity, we present this divergence in a new light. We show that
the metric acquires a logarithmic term which is the $2+1$ dimensional
realization of the Newtonian gravitational potential when General Relativity is
supplemented with a scalar field. This opens up novel possibilities which we
discuss in the conclusion.
