  In this paper, we establish existence of solutions to an indefinite coupled
non-linear system. We use partial coercivity to establish existence of a
critical point to an indefinite functional and thus the existence of solutions
on both bounded domains and doubly-periodic domains. We then take the limit of
the bounded domain to extend solutions on a bounded domain to the full-plane.
We also provide asymptotic decay analysis for solutions over the full-plane.
Necessary and sufficient conditions are also provided for the doubly periodic
domain.
