  Due to the growing capacity of gravitational-wave astronomy and black-hole
imaging, we will soon be able to emphatically decide if astrophysical objects
lurking in galactic centers are black holes. Sgr A*, one of the most prolific
astronomical radio sources in our galaxy, is the focal point for tests of
general relativity. Current mass and spin constraints predict that the central
object of the Milky Way is supermassive and slowly rotating, thus can be
conservatively modeled as a Schwarzschild black hole. The well-established
presence of accretion disks and astrophysical environments around supermassive
objects can deform their geometry and complicate their observational scientific
yield. Here, we study extreme-mass-ratio binaries comprised of a minuscule
secondary object inspiraling onto a supermassive Zipoy-Voorhees compact object;
the simplest exact solution of general relativity that describes a static,
spheroidal deformation of Schwarzschild spacetime. We examine geodesics of
prolate and oblate deformations for generic orbits and reevaluate the
non-integrability of Zipoy-Voorhees spacetime through the existence of resonant
islands in the orbital phase space. By including radiation loss with
post-Newtonian techniques, we evolve stellar-mass secondary objects around a
supermassive Zipoy-Voorhees primary and find clear imprints of
non-integrability in these systems. The peculiar structure of the primary,
allows for, not only typical single crossings of transient resonant islands,
that are well-known for non-Kerr objects, but also inspirals that traverse
through several islands, in a brief period of time, that lead to multiple
glitches in the gravitational-wave frequency evolution of the binary. The
detectability of glitches with future spaceborne detectors can, therefore,
narrow down the parameter space of exotic solutions that, otherwise, can cast
identical shadows with black holes.
