  This work concerns superharmonic perturbations of a Gaussian measure given by
a special class of positive weights in the complex plane of the form $w(z) =
\exp(-|z|^2 + U^{\mu}(z))$, where $U^{\mu}(z)$ is the logarithmic potential of
a compactly supported positive measure $\mu$. The equilibrium measure of the
corresponding weighted energy problem is shown to be supported on subharmonic
generalized quadrature domains for a large class of perturbing potentials
$U^{\mu}(z)$. It is also shown that the $2\times 2$ matrix d-bar problem for
orthogonal polynomials with respect to such weights is well-defined and has a
unique solution given explicitly by Cauchy transforms. Numerical evidence is
presented supporting a conjectured relation between the asymptotic distribution
of the zeroes of the orthogonal polynomials in a semi-classical scaling limit
and the Schwarz function of the curve bounding the support of the equilibrium
measure, extending the previously studied case of harmonic polynomial
perturbations with weights $w(z)$ supported on a compact domain.
