and China. Most of these missions require low-altitude orbits. For this reason, looking for ideal orbits that minimize or cancel the undesirable effects of some perturbations which can modify spacecraft orbits continue to be a current problem. Classically, these ideal orbits have been known as frozen orbits. A historical review of the frozen-orbit concept can be found in [1] and the references cited therein.Basically, from the mathematical point of view, a frozen orbit corresponds to an equilibrium of a system of differential equations in which the influence of the short-and medium-period terms has been removed. The reduced or double-averaged system only maintains the long-period dynamics. This system of differential equations can be derived in two ways. First, the problem can be formulated using the well-known Lagrange planetary equations, where the shortand medium-period terms can be removed by classical averaging techniques [2][3][4][5][6][7][8]. Second, the problem can also be expressed in Hamiltonian form, in which case the elimination process can be carried out by sophisticated methods based on Lie transforms [1,[9][10][11], which allow determining easily the transformations between mean and osculating elements.In general, these ideas have been employed extensively for mission-design studies. When they are applied to the case of an orbiter around the Earth, only a few zonal harmonic coefficients of the Earth's gravitational potential are enough to allow characterizing the phase-space structure of the full problem [1]. However, in the case of real mission-design analyses of low lunar orbiters, it is necessary to consider a full perturbation model which must include both a high-resolution lunar gravitational-field model and the third-body attraction caused by the Earth [12]. As a reference, Roncoli [13] recommends a 50 × 50 lunar gravitational-field model as the minimum for orbits with altitudes below 100 km, resolution later confirmed by Lara et al. in [10,11].In this context, we consider it necessary to revisit and extend the semi-analytical models developed in [14, 15], whose aim was to test the feasibility of perturbation theories based on Lie transforms, and therefore constituted mere academic proof-of-concept studies that served the author to develop specific software for this task * . Indeed, since the first high-resolution lunar gravitational fields were released [17], it is well known that high-inclination, low-altitude lunar orbits behave quite differently from what lower-degree models predict.In the present work, both the symbolic-manipulation software developed by the authors of this article and our expertise from previous models are applied to the analysis of an orbiter in a low-altitude near-circular orbit around the Moon, perturbed by the combined effect of a 50 × 50 lunar gravitational-field model and the third-body attraction caused by the Earth. Our objective is to develop a semi-analytical theory that can be integrated both in a symbolic software tool for the computation of real low-altitud...