  Non-selfgravitating equilibrium tori orbiting around black holes have a long
history and have been employed in numerous simulations of accretion flows onto
black holes and other compact objects. We have revisited the problem of
constructing such equilibria starting from spherically symmetric black-hole
spacetimes expressed in terms of a fully generic and rapidly converging
parameterisation: the RZ metric. Within this framework, we have extended the
definitions of all of the quantities characterising these equilibria, starting
from the concept of the von Zeipel cylinders and up to the possible ranges of
the specific angular momenta that are employed to construct families of tori.
Within the allowed space of parameters we have then encountered both standard
``single-torus'' solutions, but also non-standard ``double-tori'' solutions.
While the properties of the first ones in terms of the presence of a single
cusp, of a local pressure maximum and of a varying outer radius, are very
similar to those encountered in general relativity, the properties of
double-tori solutions are far richer and naturally allow for configurations
having the same constant specific angular momentum and hence are potentially
easier to produce in nature. The existence of these objects is at present very
hypothetical, but these equilibrium tori were to be observed, they would
provide very valuable information on the properties of the spacetime and on its
deviation from general relativity.
