  In this paper we extend vessel theory, or equivalently, the theory of
overdetermined $2D$ systems to the Pontryagin space setting. We focus on
realization theorems of the various characteristic functions associated to such
vessels. In particular, we develop an indefinite version of de Branges-Rovnyak
theory over real compact Riemann surfaces. To do so, we use the theory of
contractions in Pontryagin spaces and the theory of analytic kernels with a
finite number of negative squares. Finally, we utilize the indefinite de
Branges-Rovnyak theory on compact Riemann surfaces in order to prove a Beurling
type theorem on indefinite Hardy spaces on finite bordered Riemann surfaces.
