A coupling normal form for reversible center-center Hamiltonians with application to the restricted three-body problem

Near a reversible center–center equilibrium, the Birkhoff normal form does not describe the families bifurcating from the continued tori, while resonant normal forms capture these bifurcations through a reduced system. For such equilibria with a reflection symmetry, we develop a coupling normal form in which the bifurcating families also appear as tori of action normal forms. Along a selected coupling direction, the first slow harmonic is absorbed into an action-dependent counterterm. We prove that the coupled recursion determines unique even counterterm coefficients, a formal action normal form, and an exact counterterm factorization. The factorization gives a scalar recovery constraint, whose trivial factor corresponds to the Birkhoff normal form. At every finite order, a symmetric pair of bifurcating branches leaves the trivial branch at a critical curve and carries approximately invariant tori and circles with explicit residual estimates. Applied to the spatial circular restricted three-body problem, the construction yields approximate halo orbits and quasi-halo tori.