  We study several classes of holomorphic dynamical systems associated with
quadrature domains. Our main result is that real-symmetric polynomials in the
principal hyperbolic component of the Mandelbrot set can be conformally mated
with a congruence subgroup of $\textrm{PSL}(2,\mathbb{Z})$, and that this
conformal mating is the Schwarz function of a simply connected quadrature
domain.
