  In this paper, we prove \emph{a priori} estimates for some vortex-type
equations on compact Riemann surfaces. As applications, we recover existing
estimates for the vortex bundle Monge-Amp\`ere equation, prove an existence and
uniqueness theorem for the Calabi-Yang-Mills equations on vortex bundles, and
get estimates for $J-$vortex equation. We prove an existence and uniqueness
result relating Gieseker stability and the existence of almost Hermitian
Einstein metrics, i.e., a Kobayashi-Hitchin type correspondence. We also prove
K\"ahlerness of the negative of the symplectic form which arises in the moment
map interpretation of the Calabi-Yang-Mills equations in \cite{Vamsi3}
