Bum-Puzzled-U-2] found a similar solution. It was first considered by Domb who considered all 2-suns with one pair of poles. The general solution of this system contains two types of orbits, the tadpole and the loop. The tadpole orbits are periodic and describe circular orbits. The tadpole orbits can contain different numbers of loops. The loop orbits also have a periodic character but their size (the diameter of the loop) is determined by the parameters of the system. A characteristic feature of these orbits is that the smaller loop is contained in the bigger one. We call this orbit configuration one [*loop*]{}. Thus, we can see that it is possible to combine one loop with the tadpole orbits. For example, on the tadpole orbit without a loop we can put the one loop at one pole. Thus, we can see that the tadpole orbit plus a loop orbit can describe all orbits in Domb’s solution. The two solutions were presented in Domb’s paper[@Domb-Dim-Aver-2] but it was not easy to understand the nature of these orbits and all possible combinations of orbits.
The other possibility is to construct all periodic orbits with one line of symmetry is two solutions by Domb for 2-sun systems with one pair of lines of symmetry as shown in [@Bum-Puzzled-U-2].
The two orbits for the second case are $$\begin{array}{rcl}
\mathbf{z}^{T}&=&\left(-1,\frac{1}{2},\frac{1}{2},\frac{1}{2},0\right),\\
\mathbf{z}^{T}&=&\left(-1,\frac{1}{2},\frac{1}{2},-\frac{1}{2},0\right).\\
\end{array}$$
In this case there are three solutions for the two-suns with two lines of symmetry[@Bum-Puzzled-U-2]. These orbits are shown in Figure \[fig5\].
![\[fig5\]Orbits for 2-sun system with two lines of symmetry. ](graph_orbits2sun_with_lines_of_symmetry_v4.jpg){width="80.00000%"}
The first orbit is a loop with one pair of poles. The second orbit is an orbit that can be obtained from the loop orbit by the transformation $\varphi\to\pi-\varphi$ and the third orbit is a tadpole orbit without a loop.
All 3 orbits can be combined into one orbit. The orbit in Figure \[fig5\] (d) is obtained from the loop orbit in Figure \[fig5\] (b) by adding a line of symmetry to the outer loop. The second orbit (Figure \[fig5\] (c)) is obtained from the tadpole orbit in Figure \[fig5\] (a) by adding a line of symmetry to the middle.
All possible orbits for the three-suns system have the following form[@Bum-Puzzled-U-2] $$\begin{array}{rcl}
\mathbf{z}^{T}&=&\left(m,\frac{1}{2}+m,\frac{1}{2}+m,\frac{1}{2}+m,n\right),\\
\mathbf{z}^{T}&=&\left(m,\frac{1}{2}+m,-\frac{1}{2}-m,\frac{1}{2}+m,n\right),\\
\mathbf{z}^{T}&=&\left(m,\frac{1}{2}+m,-\frac{1}{2}-m,-\frac{1}{2}-m,n\right),\\
\end{array}$$ where $m,n=0,\pm1$. The orbit with $m=0$ and $n=1$ is a tadpole orbit, the orbit with $m=0$ and $n=-1$ is a loop orbit. All other orbits have one or two loops.
![All orbits for the three-suns system[@Bum-Puzzled-U-2] with one or two lines of symmetry.[]{data-label="fig6"}](3sun_graph.jpg){width="70.00000%"}
Using the formulas for the tadpole orbit and the loops and the transformation $S_{2}^{n}$ from Section \[sec:symm\] we can construct orbits with one line of symmetry and one loop from the orbits in Figure \[fig6\] and vice versa.
Figure \[fig6\] shows all orbits for the three-suns system with one or two lines of symmetry.
In [@Bum-Puzzled-U-2] we can see that in this way we can get a large number of orbits (we only consider systems with one pair of lines of symmetry). Orbits for systems with two pairs of lines of symmetry can be obtained by using these orbits as seeds. Also, the transformation $\varphi\to\pi-\varphi$ describes the symmetry with respect to the x-y plane.
We have now a collection of formulas to calculate orbits for different $N$-sun systems. We can see the symmetry of the orbits with respect to $180^{\circ}$ rotations. From the symmetry of these orbits and the formulas for the distance of the points of an orbit and the angles between points we see that for each orbit $\varphi$ we have to get a subset of points to get an orbit. As it was shown above for $N$-suns with one line of symmetry we need 2(N-1) parameters. Now it is easy to see that for the $N$-suns with one line of symmetry and one loop we need 6(N-2) parameters.
All possible types of orbits that can exist for $N$-suns can be easily deduced by the rules described above from the possible orbits that can exist for two-suns. We note that a detailed discussion of orbits for $N$-suns with two pairs of lines of symmetry is given in [@Bum-Puzzled-U-2].
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