In this paper, as a continuation of [Fernandez-Guasti, \textit{Celest Mech
Dyn Astron} 137, 4 (2025)], we demonstrate the maximal superintegrability of
the reduced Hamiltonian, which governs the four-body choreographic planar
motion along the lima\c{c}on trisectrix (resembling a folded figure eight), in
the six-dimensional space of relative motion. The corresponding eleven
integrals of motion in the Liouville-Arnold sense are presented explicitly.
Specifically, it is shown that the reduced Hamiltonian admits complete
separation of variables in Jacobi-like variables. The emergence of this
choreography is not a direct consequence of maximal superintegrability. Rather,
it originates from the existence of \textit{particular integrals} and the
phenomenon of \textit{particular involution}. The fragmentation of a more
general four-body choreographic motion into two isomorphic two-body
choreographies is discussed in detail. This model combines choreographic motion
with maximal superintegrability, a seldom-studied interplay in classical
mechanics.

Cet article explore les excursions dans le temps et leurs implications.

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