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Some numbers appear with opposite signs compared to Andreu (1998) due to the coordinate change.Table 3 Coefficients for the QBCP: \(\alpha _{1j}, \alpha _{2j}, \alpha _{3j}\), recreated from Andreu (1998)Full size tableTable 4 Coefficients for the QBCP: \(\alpha _{4j}, \alpha _{5j}, \alpha _{6j}\), recreated from Andreu (1998)Full size tableTable 5 Coefficients for the QBCP: \(\alpha _{7j}, \alpha _{8j}\), recreated from Andreu (1998)Full size tableAppendix B 2D-QPO computation algorithmWhile multiple numerical schemes exist to approximate the QPO structures, an algorithm developed by Gómez and Mondelo (2001) as well as Olikara and Scheeres (2012) (GMOS) is leveraged. Other related references include Olikara and Howell (2010), Olikara et al. (2016), Baresi et al. (2018), McCarthy and Howell (2021). The GMOS algorithm formulates a targeting problem to numerically compute a 2D-QPO by leveraging a stroboscopic map with the period corresponding to one of the frequencies. While either the latitudinal frequency, \(\nu _{lat}\), or the longitudinal frequency, \(\nu _{lon}\), may be utilized, \(\nu _{lon}\) is selected for a consistent analysis in the current investigation. Then, at the stroboscopic map, the 2D-QPO now appears as an invariant curve, governed by the latitudinal angle, \(\theta _{lat}\). Denoting the invariant curve as \(\vec {u} = \vec {u}(\theta _{lat})\), the discrete Fourier transform (DFT) is leveraged to numerically approximate the smooth invariant curve with finite sample points. Thus,$$\begin{aligned} \vec {u}(\theta _{lat}) \approx \sum _{k=0}^{N-1} \vec {D}_k e^{\left( j\theta _{lat} k \right) }, \end{aligned}$$ (B3) where N is the number of sample points, assumed to be 25 in the current investigation and \(j = \sqrt{-1}\). The Fourier coefficients are denoted as \(\vec {D}\). The N selected points along the invariant curve are propagated for one stroboscopic mapping time, i.e., the longitudinal period. The end state is denoted as \(\vec {\Psi }(\vec {u}(\theta _k)) \), \(1 \le k \le N\), where \(\vec {\Psi }\) is the stroboscopic mapping function under a selected dynamical model. These end states are also on the invariant curve but rotate by \(\sigma \), the rotation number. Thus,$$\begin{aligned} \vec {\Psi }(\vec {u}(\theta _{lat})) = \vec {u}(\theta _{lat} + \sigma ), \end{aligned}$$ (B4) resulting in the invariance condition. This condition is leveraged to formulate a numerical targeting problem that solves for 2D-QPOs.Appendix C Eccentricity from Hill’s lunar theoryFrom Darwin (2009/1916), in the vicinity of the lunar variational PO, the linear approximation for the in-plane invariant curve constructed at \(\theta _2 = 0\) is expressed in terms of the approximate eccentricity, \(e_{a}\), as follows,$$\begin{aligned}&\xi (\theta _2 = 0)\approx A_0\left( 1-m^2 -\frac{15}{8} m e_{a}\cos ( \theta _1) - e_{a}\cos (\theta _1)\right) \end{aligned}$$ (C5) $$\begin{aligned}&\eta (\theta _2 = 0) \approx A_0\left( -\frac{15}{4} m e_{a}\sin (\theta _1) + 2e_{a}\sin (\theta _1) \right) , \end{aligned}$$ (C6) where \(A_0 = a_0 (1-m^2/6)\) leveraging \(a_0\)