  Recent gravitational wave observations and shadow imaging have demonstrated
the astonishing consistency of the Kerr paradigm despite all the special
symmetries assumed in deriving the Kerr metric. Hence, it is crucial to test
the presence of these symmetries in astrophysical scenarios and constraint
possible deviations from them, especially in strong field regimes. With this
motivation, the present work aims to investigate the theoretical consequences
and observational signatures of non-circularity in a unified theory-agnostic
manner. For this purpose, we construct a general non-circular metric with a
small parameterized deviation from Kerr. This metric preserves the other
properties of Kerr, such as stationarity, axisymmetry, asymptotic flatness, and
the equatorial reflection symmetry. Apart from the resulting mathematical
simplifications, this assumption is crucial to disentangle the consequences of
relaxing circularity from other properties. Then, after discussing various
novel theoretical consequences, we perform a detailed analysis of extreme mass
ratio inspirals and Lense-Thirring precession in the context of this newly
constructed metric. Our study clearly shows the promising prospects of
detecting and constraining even a slight non-circular deviation from the Kerr
paradigm using the future gravitational wave observations by the Laser
Interferometer Space Antenna.
