  According to general relativity, an isolated black hole in vacuum shall be
described by the Kerr metric, whose geodesic equations are integrable. The
violation of integrability leads to chaos for particles moving around the black
hole. This chaotic dynamics could leave imprints on the associated
gravitational waveform and could be tested with upcoming observations. In this
paper, we investigate the chaotic orbital dynamics induced by the violation of
a certain spacetime symmetry, the circularity. Specifically, we focus on the
resonant orbits of a particular noncircular spacetime as an example and find
that they form chains of Birkhoff islands on Poincar\'e surfaces of section. We
compare the island structures with those generated in typical nonintegrable but
circular spacetimes. The islands of stability induced by noncircularity appear
asymmetric on the most common Poincar\'e surface of section at the equatorial
plane. The asymmetric patterns of islands vary discontinuously when the
spacetime parameters transit through integrable regions. The origin of such
features is explained in the context of perturbation analysis by considering
the orbits associated with stable fixed points on the section. Possible
observational implications about testing circularity through gravitational wave
detection are discussed.
