  The aim of this paper is to inter-relate several algebraic and analytic
objects, such as real-type algebraic curves, quadrature domains, functions on
them and rational matrix functions with special properties, and some objects
from Operator Theory, such as vector Toeplitz operators and subnormal
operators. Our tools come from operator theory, but some of our results have
purely algebraic formulation. We make use of Xia's theory of subnormal
operators and of the previous results by the author in this direction. We also
correct (in Section 5) some inaccuracies in two papers by the author in Revista
Matematica Iberoamericana (1998).
