Foliation on the moduli space of extrinsic circular trajectories on a complex hyperbolic space

We show that every circle on a complex hyperbolic space except unbounded circles of complex torsion ±1 which have two distinct points at infinity can be regarded as a trajectory for some Sasakian magnetic field on some totally η-umbilic real hypersurface. Our study explains why the lamination on the moduli space of circles on a complex hyperbolic space has singularities only on the leaf of circles of complex torsion ±1.