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.

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