  It is well known that, in the plane, the boundary of any quadrature domain
(in the classical sense) coincides with the zero set of a polynomial. We show,
by explicitly constructing some four-dimensional examples, that this is not
always the case. This confirms, in dimension 4, a conjecture of the second
author. Our method is based on the Schwarz potential and involves elliptic
integrals of the third kind.
