  In this article, we identify the 5-dimensional analogue of the finite energy
foliations introduced by Hofer--Wysocki--Zehnder for the study of 3-dimensional
Reeb flows, and show that these exist for the spatial circular restricted
three-body problem (SCR3BP) whenever the planar dynamics is convex. We
introduce the notion of a fiberwise-recurrent point, which may be thought of as
a symplectic version of the leafwise intersections introduced by Moser, and
show that they exist in abundance for a perturbative regime in the SCR3BP. We
then use this foliation to induce a Reeb flow on the standard 3-sphere, via the
use of pseudo-holomorphic curves, to be understood as the best approximation of
the given dynamics that preserves the foliation. We discuss examples, further
geometric structures, and speculate on possible applications.
