In this paper we investigate the planar circular restricted ( \(3 + 1\) )-body problem. A massless particle moves under the Newtonian gravitational influence of three primaries. These primaries revolve in circular orbits around their common center of mass within the same plane, forming a collinear configuration. We establish the existence and nonlinear stability of equilibrium points under the condition that two of the primaries have equal masses, denoted by \(\mu \) . Through analytical and numerical methods, we identify a critical mass parameter \(\mu _1 \approx 0.0743\) where a Hamiltonian–Hopf bifurcation occurs at non-collinear equilibrium points. Furthermore, we prove the existence of Hill-type periodic orbits and KAM tori surrounding these periodic orbits in this circular restricted four-body problem.
