  We study geometrically thick perfect-fluid tori with constant specific
angular momentum, so-called "Polish doughnuts", orbiting deformed compact
objects with a quadrupole moment. More specifically, we consider two different
asymptotically flat, static and axisymmetric vacuum solutions to Einstein's
field equation with a non-zero quadrupole moment, the q-metric and the
Erez-Rosen spacetime. It is our main goal to find features of Polish doughnuts
in these two spacetimes which qualitatively distinguish them from Polish
doughnuts in the Schwarzschild spacetime. As a main result we find that, for
both metrics, there is a range of positive (Geroch-Hansen) quadrupole moments
which allows for the existence of double tori. If these double tori fill their
Roche lobes completely, their meridional cross-section has the shape of a fish,
with the body of the fish corresponding to the outer torus and the fish-tail
corresponding to the inner torus. Such double tori do not exist in the
Schwarzschild spacetime.
