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We derive new closed form analytic solutions for general non-equatorial eccentric bound trajectories, $\{ \phi \left( r, \theta \right)$, $\ t \left( r, \theta \right),\ r \left( \theta \right) \}$, around a Kerr black hole by using the transformation $1/r=\mu \left(1+ e \cos \chi \right)$. We obtain and implement translation relations between energy and angular momentum of the particle, ($E$, $L$), and eccentricity and inverse-latus rectum, ($e$, $\mu$), for a given spin, $a$, and $Q$ to obtain the trajectory completely in the ($e$, $\mu$, $a$, $Q$) parameter space. The solutions for the specific case of non-equatorial separatrix orbits uses the $\chi$ variable, which has direct applications to gravitational wave astronomy. Using long time averages, we derive the novel closed form expressions of the fundamental frequencies of non-equatorial eccentric trajectories, ($\nu_{\phi}$, $\nu_{\theta}$, $\nu_{r}$). We discuss simple reduced formulae for non-equatorial eccentric, non-equatorial separatrix, zoom whirl, spherical orbits, and their importance in selected astrophysical applications.

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We study equatorial eccentric orbits in Kerr geometry and apply relativistic precession model to the cases of black hole X-ray binaries (BHXRB) where two HFQPOs (High-Frequency Quasi-Periodic Oscillations) and Type-C QPO are simultaneously observed, or to the cases where only two simultaneous HFQPOs are observed, for example in GROJ1655-40, M82X-1, XTEJ1550-564, 4U1630-47, and GRS1915+105. We assume that the frequencies to be Gaussian or Gamma distributed and calculate probability distributions to infer eccentricity $e$, periastron distance $r_p$, of the orbit giving rise to QPOs, and the spin of the black hole, $a$. In the equatorial case, we find that the eccentric solutions are found in the region between ISCO and the marginally bound orbit. We also establish the primacy of eccentric orbits in this region for the generation of 3:2 ($\mathcal{R}=\nu_{\phi}/(\nu_{\phi}-\nu_{r}) \simeq1.5$) commensurability of QPOs as a strong possibility. We also explore the dynamical parameter space to produce commonly found fundamental frequency ratios for spherical ($Q\neq0$) bound trajectories in the coronal region.

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