  The detection of gravitational waves from extreme-mass-ratio inspirals with
upcoming space-borne detectors will allow for unprecedented tests of general
relativity in the strong-field regime. Aside from assessing whether black holes
are unequivocally described by the Kerr metric, such detections may place
constraints on the degree of spacetime symmetry. In particular, depending on
exactly how a hypothetical departure from the Kerr metric manifests, the Carter
symmetry, which implies the integrability of the geodesic equations, may be
broken. Here, we examine the gravitational waveforms associated with
non-integrable extreme-mass-ratio inspirals involving a small-mass companion
and a supermassive compact object of general relativity, namely the
Manko-Novikov spacetime. We show that the waveforms displays sudden frequency
jumps, when the companion transverses resonant islands. These findings
demonstrate that such abrupt manifestations in the gravitational-wave
frequencies are generic, have a genuine astrophysical origin and function as a
distinctive signature of chaotic phenomena in extreme-mass-ratio binaries.
