  We study the Abelian Higgs vortex solutions to the sinh-Gordon equation and
the elliptic Tzitzeica equation. Starting from these particular vortices, we
construct solutions to the Taubes equation with higher vortex number, on
surfaces with conical singularities.
  We then, analyse more general properties of vortices on such singular
surfaces and propose a method to obtain vortices on conifolds from vortices on
surfaces of revolution. We apply our method to construct explicit vortex
solutions on the Poincar\'e disk with a conical singularity in the centre, to
which we refer as the "hyperbolic cone".
  We uplift the Abelian sinh-Gordon and Tzitzeica vortex solutions to four
dimensions and construct cylindrically symmetric, self-dual Yang-Mills
instantons on a non-self-dual (nor anti-self-dual) $4$-dimensional K\"ahler
manifold with non-vanishing scalar curvature. The instantons we construct in
this way cannot be obtained via a twistorial approach.
