  The post-Newtonian orbital effects induced by the mass quadrupole and spin
octupole moments of an isolated, oblate spheroid of constant density that is
rigidly and uniformly rotating on the motion of a test particle are
analytically worked out for an arbitrary orbital configuration and without any
preferred orientation of the body's spin axis. The resulting expressions are
specialized to the cases of (a) equatorial and (b) polar orbits. The
opportunity offered by a hypothetical new spacecraft moving around Jupiter
along a Juno-like highly elliptical, polar orbit to measure them is
preliminarily studied. Although more difficult to be practically implemented,
also the case of a less elliptical orbit is considered since it yields much
larger figures for the relativistic effects of interest. The possibility of
using the S stars orbiting the supermassive black hole in Sgr A$^\ast$ at the
Galactic Center as probes to potentially constrain some parameters of the
predicted extended mass distribution surrounding the hole by means of the
aforementioned orbital effects is briefly examined.
