## 1 Introduction
[PERSON], who gave in [6] the initial algorithmic definition of a _weak stability boundary_ (WSB) as a boundary of the region in the configuration space where the orbital motion around one of the celestial bodies is stable, later formalized this concept: the weak stability boundary of a given body is a set in the phase space that is located in the intersection of a certain Jacobi integral manifold and the region bounded by the hypersurface of zero Keplerian energy [8; 9]. In contrast to capture by executing an LOI maneuver, ballistic capture is temporary. It is also referred to as weak. To design a low-energy transfer trajectory in the three-body (Earth-Moon) or four-body (Earth-Moon-Sun) system, many numerical techniques have been developed that are based on searching for heteroclinic connections between the invariant manifolds of (quasi)periodic orbits around the \(L_{1}\) and/or \(L_{2}\) libration points [10; 11; 12; 13]. In the planar CR3 BP, using the tools of Poincare and Keplerian maps, researchers have revealed the chaotic nature of the ballistic capture phenomenon and its intrinsic link with both the WSB and resonant orbits. Moreover, the _resonance hopping theory_ has been developed [8; 9; 14; 15; 16; 17]; convincing evidence has been discovered that there exists a close connection between the WSB set and invariant manifolds of libration point orbits [18; 19]. When aerospace engineers refer to a WSB trajectory, they mean a Hiten-like low-energy trajectory starting from near-Earth orbit and exploiting the solar gravitational perturbation to ensure exterior ballistic capture by the Moon. Such a trajectory was utilized in NASA's GRAIL mission in the period 2011-2012 [20] and, more recently, in the missions of CAPSTONE [21] and Danuri [22]. Depending on the final lunar orbit, a transfer trajectory may or may not include a large LOI impulse. In the latter case, a transfer can be reasonably called _ballistic_, though the term of _ballistic lunar transfer_ (BLT) is often applied to any trajectory of this sort, no matter the magnitude of the required LOI impulse. Moreover, in our opinion, neither the name WSB nor BLT reflects the gravity of the Sun's effect, the key feature of such transfer trajectories that singles them out from other low-energy transfers. So, we would like to coin the term of _Sun-assisted lunar transfer_ (SALT). A SALT trajectory may be viewed as a special case of the broader class of WSB trajectories (for example, it is natural to consider SMART-like transfer trajectories to be included in the WSB class as well). SALT trajectories can be ballistic or thrust-augmented, as in the cases of recently launched Lunar IceCube [23] and EQUULEUS [24] missions. The simplest dynamical model capturing the crucial Sun's gravity perturbation effect is the bicircular restricted four-body problem (BR4 BP), an extension of CR3 BP where the Sun is also assumed to revolve in a distant circular orbit around the Earth-Moon barycenter. As the extensive global search for optimal two-impulse transfers in the planar BR4 BP model clearly demonstrates, it is SALT trajectories that achieve the maximum fuel efficiency, especially if an intermediate lunar flyby is employed [25; 26; 27]. Intensive research on the development and application of SALT design techniques is now actively ongoing, primarily with periodic and quasiperiodic lunar libration point orbits as a transfer destination [28; 29; 30; 31; 32; 33; 34]. To this moment, all the existing techniques of designing a SALT trajectory rely on direct numerical optimization procedures. When possible, the invariant manifold geometry can be employed (e.g., by using the Poincare periapsis map) to obtain a good initial guess for such a procedure. The major obstacle to the regular convergence of the procedure from some initial guess is due to the fact that the initial and final legs of a lunar transfer trajectory obey the fast near-Earth/near-Moon dynamics, which causes extreme sensitivity to boundary conditions.