that represent the families of SPO in the initial conditions map, in this case, calculated using the Grid Search Method [27].

[Figure: Fig. 11: 3-D polyhedral shape model of the nucleus of comet 67P viewed from xOz plane, rotating 90◦ around the x−axis and 340◦around the z−axis. The color box indicates the surface distance from the center of mass (in km). The two black vertical lines indicate the considered polyhedral solid region for the comet’s neck ∼1 km (-0.2 to 0.8 km).]

For comet 67P, using m = 1, mtol = 10−8, and P = 30 periods, 12 SPO families were found, numbered from 1 to 12 on the initial conditions map, each represented by a color according to the legend (Fig. 12). A point on the characteristic curve represents each orbit of the family. Table 5 presents the topological classification of the characteristic curves that enables the identification of characteristics of orbital families without the need for direct visualization of the orbits [38]. It shows the common features shared by all orbits within a family, as well as the variations occurring at the endpoints of each curve, based on the points encircled by the orbit, namely, the central body (denoted by DS) and the equilibrium points, whose locations and designations correspond to those in Fig. 7, with L2 and L4 being the triangular points collectively represented by the single letter T. This classification also accounts for the direction of motion: “Direct”, denoted by D( ), for cases where x0 > xP/2, with xP/2 defined as the x-position of the period P/2 where the curve intersects the x−axis perpendicularly, thereby characterizing a SPO; and “Retrograde”, denoted by R( ), for cases where x0 < xP/2. Consequently, the notation [ ] signifies that all orbits within the family exhibit similar characteristics; [→...] indicates that the orbits at one endpoint conform to the common family characterization while the other endpoint are different; and [... →...] implies that both endpoints differ from the common characterization. Note, in Tab. 5, that regions and corresponding families organize it, and the common part of the representation is highlighted in bold. Based on this classification and in the Figs. 13, 14, 15, some analyses can be conducted:
- Family 1 is the only one that covers two zones (Zl and Zd), covering a wide range of the Jacobi constant. It represents elliptical curves around the body, which tend to be almost circular and possibly circular (Fig. 13 (a)).

[Figure: Fig. 12: Map of the characteristic curves of comet 67P. On the abscissa axis, we have the initial position x0, and on the ordinate axis, the Jacobi constant J. Zl, Zr, and Zd name the quadrants where there are solutions. The colors of the lines represent each SPO family. The gray area represents the forbidden region, where for y = 0, the condition J > −(x2 + y2) + 1]

2U indicating a negative particle speed. The lower left figure is a closer view of the A rectangle in the Zl zone, and the lower right figure is a closer view of the B rectangle in the Zr zone.
- From Family 2 to Family 5, the curves are in the Zl zone where x0 < 0, fulfilling the requirement of x0 < xP/2. In these cases, the orbits are retrograde. Some orbits of these families are represented in (Fig. 13 (b), 13 (c), 13 (d) and 14 (a)).