  The Abelian Higgs model on a compact Riemann surface \Sigma of genus g is
considered. We show that for g > 1 the Bogomolny equations for multi-vortices
at critical coupling can be obtained as compatibility conditions of two linear
equations (Lax pair) which are written down explicitly. These vortices
correspond precisely to SO(3)-symmetric Yang-Mills instantons on the
(conformal) gravitational instanton \Sigma\times S^2 with a scalar-flat Kahler
metric. Thus, the standard methods of constructing solutions and studying their
properties by using Lax pairs (twistor approach, dressing method etc.) can be
applied to the vortex equations on \Sigma. In the twistor description,
solutions of the integrable vortex equations correspond to rank-2 holomorphic
vector bundles over the complex 3-dimensional twistor space of \Sigma\times
S^2. We show that in the general (nonintegrable) case there is a bijection
between the moduli spaces of solutions to vortex equations on \Sigma and of
pseudo-holomorphic bundles over the almost complex twistor space.
