Particles with precessing spin in Kerr spacetime: Analytic solutions for eccentric orbits and homoclinic motion near the equatorial plane

We present a family of analytic solutions for the nearly-equatorial motion of a test particle with precessing spin in Kerr spacetime. We solve the equations of motion up to linear order in the small body's spin for periodic and homoclinic orbits. At zero order, the particle moves along equatorial geodesics. The spin-curvature force introduces post-geodesic corrections which, for generic spin orientations, cause the precession of the orbital plane. We derive the solutions for eccentric orbits in terms of Legendre elliptic integrals and Jacobi elliptic functions for generic referential geodesics (known as ``spin gauges"). Our analytical solutions perfectly match the numerical trajectories obtained by Drummond and Hughes in Phys. Rev. D 105, 124041 (2022), and Piovano et al. in Phys. Rev. D 111, 044009 (2025). Furthermore, we present, for the first time, the solutions for homoclinic orbits for a spinning particle in Kerr spacetime, and the spin-corrections to the location of the separatrix. The homoclinic trajectories are described in closed form using elementary functions. Finally, we introduce a novel parametrization for the motion of a spinning particle, called ``fixed eccentricity spin gauge". This is the only spin gauge in which the corrections to periodic orbits are finite at the geodesic separatrix, and continuously reduce to the last stable orbits under appropriate limits. Our results will be useful for modeling the inspiral and transition-to-plunge phases of asymmetric mass binaries within the two-time-scale framework.