## 5 Conclusions  
The considered approach based on the use of a flat auxiliary problem, has shown its effectiveness in solving the problem of designing a low-energy trajectory. At the same time, it is important to note that the \"another one dimension\" of the main problem does not introduce any additional sufficient complexities in a solving process. Because the configuration of celestial bodies that was derived as a solution from planar auxiliary problem, provides at the same time the monotonically changing of declination between SC osculating trajectory and Moon orbital plane and thus with sufficiently high rate of change. Alternatively, we can say that the Sun influence provide a rapid alignment between Moon orbital plane and SC trajectory for an optimal configuration from auxiliary solution. By itself, it is a well-known fact and it can easily be checked in a different works [1; 2; 5; 14]. The main advantage in a proposed approach in comparison with the other works [1; 2; 3; 4; 5; 6; 7; 8; 9; 10] is an ability to reduce complexity of solving _main_ problem, cause all other known approaches offers to directly solve problem like (29) (which is very sensitive and multimodal) with or without any suitable initial guess. The latter is not depends on a form of analytical conditions or constraints that should be used in other approaches. The usage of the auxiliary problem help us to build the admissible initial approach for solving problem (29). Any subproblem from the 3-stage approach can be solved numerically sufficiently easy with any local constraint optimization algorithm. It makes the proposed approach some kind more \"engineering\". The auxiliary problem has a greater numerical stability in comparison with other--due to its simplified \"nature\" of cause, but also, due to usage of the system like (13), whose state vector consists of osculating elements. In addition, for some other problems (4 BP) with a possible multirevolutional SC motion around the main body, system (13) can be used in a combined manner with asymptotical technic (numerical or maybe analytical averaging). Conceptualization, I.N. and V.S. ; methodology, I.N. ; software I.N. and V.S. ; validation, I.N. and V.S. ; formal analysis, I.N. ; investigation, I.N. and V.S. ; resources, V.S. ; data curation, V.S. ; writing--original draft preparation, I.N. ; writing--review and editing I.N. ; visualization, V.S. ; supervision, I.N. ; project administration, I.N. ; funding acquisition, I.N. All authors have read and agreed to the published version of the manuscript. This work was supported by the Russian Science Foundation grant no. 22-79-10206, [[https://rscf.ru/project/22-79-10206/](https://rscf.ru/project/22-79-10206/)]([https://rscf.ru/project/22-79-10206/](https://rscf.ru/project/22-79-10206/)), accessed on 1 August 2022. Not applicable. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\mathbf{N}\) & \(\mathbf{\Delta V_{0}}\), [km/s] & \(\mathbf{\Delta V_{2}}\), [km/s] & \(\mathbf{\Delta V_{t}}\), [km/s] & \(\mathbf{\Delta V^{\Sigma}}\), [km/s] & **Start Date** & \(\mathbf{T_{full}}\), [days] \\ \hline
1 & 3.19624 & 0 & 0.64323 & 3.83745 & 21 January 2027 & 105.348 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The optimal solution of the auxiliary problem (25). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\mathbf{N}\) & \(\mathbf{\Delta V_{0}}\), [km/s] & \(\mathbf{\Delta V_{t}}\), [km/s] & \(\mathbf{\Delta V^{\Sigma}}\), [km/s] & \(\mathbf{T_{full}}\), [days] \\ \hline
1 & 3.19624 & 0.64323 & 3.83745 & 105.348 \\ \hline
2 & 3.19445 & 0.63476 & 3.82922 & 103.579 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The comparison between solutions of the main problem (bold) and the auxiliary problem.