The class of five integrable vortex equations discussed recently by Manton is
extended so it includes the relativistic BPS Chern-Simons vortices, yielding a
total of nineteen vortex equations. Not all the nineteen vortex equations are
integrable, but four new integrable equations are discovered and we generalize
them to infinitely many sets of four integrable vortex equations, with each set
denoted by its integer order $n$. Their integrability is similar to the known
cases, but give rise to different (generalized) Baptista geometries, where the
Baptista metric is a conformal rescaling of the background metric by the Higgs
field. In particular, the Baptista manifolds have conical singularities. Where
the Jackiw-Pi, Taubes, Popov and Ambj{\o}rn-Olesen vortices have conical
deficits of $2\pi$ at each vortex zero in their Baptista manifolds, the
higher-order generalizations of these equations are also integrable with larger
constant curvatures and a $2\pi n$ conical deficit at each vortex zero. We then
generalize a superposition law, known for Taubes vortices of how to add
vortices to a known solution, to all the integrable vortex equations. We find
that although the Taubes and the Popov equations relate to themselves, the
Ambj{\o}rn-Olesen and Jackiw-Pi vortices are added by using the Baptista metric
and the Popov equation. Finally, we find many further relations between vortex
equations, e.g. we find that the Chern-Simons vortices can be interpreted as
Taubes vortices on the Baptista manifold of their own solution.
