where r$_{min}$ and r$_{max}$ are the real positive solutions of the equation We find after some calculation where sin α = Jc κ 2 ( 3 κ 2 2 E ) 3 $^{2}$, and 0 ≤ α ≤ π $_{2}$. We now return to Eq. (2.8). An immediate consequence of integration of Eq. (2.8) is the the relation 2 In the particular case of r$_{0}$ = r$_{min}$ = 0, and r ( t ) > r$_{min}$ , t > 0, the integral from the right-hand side of (2.16) can be expressed by means of the elliptic integral of the third kind Π( φ, n, k ) (see Ref. [19], 3.137, Eq. 3), namely we have where $^{r}$− is the negative root of the polynomial from the left-hand side of Eq. (2.13) satisfying Clearly, Eq. (2.17) defines r as an implicit function of ϕ . Furthermore, for r$_{0}$ = r$_{max}$ = 0, and r ( t ) < r$_{max}$ , t > 0, the implicit equation for the trajectory can be obtained from (2.16) with the help of the elliptic functions of the third kind Π( φ, n, k ) and first kind F ( φ, k ) (see [19], 3.137, Eq. 4). It follows that 5 ϕ = ϕ$_{0}$ − 2 Jc κ $^{2}$r 2 − r 2 max √ r 2 − − r 2 min [ ( r 2 − − r 2 $_{max}$) × Π ( arcsin √ ( r 2 − − r 2 $_{min}$)( r 2 max − r $^{2}$) ( r 2 max r 2 $_{min}$)( r 2 − r $^{2}$) , r 2 max − r 2 min r 2 − r 2 min r 2 − r 2 max , √ r 2 max − r 2 min r 2 − r 2 min ) + r 2 $_{max}$F ( arcsin √ ( r 2 − − r 2 $_{min}$)( r 2 max − r $^{2}$) ( r 2 max r 2 $_{min}$)( r 2 − r $^{2}$) , √ r 2 max − r 2 min r 2 − r 2 min ) ] , r$_{0}$ = r$_{max}$. (2.19) r$_{min}$ + r$_{max}$ + $^{r}$− = 0 , r 2 − > r 2 max > r 2 $_{min}$. (2.18) ϕ = ϕ$_{0}$ + 2 Jc κ $^{2}$r 2 min √ r 2 − − r 2 min Π ( arcsin √ r 2 − r 2 min r 2 max r 2 min , 1 − r 2 max r 2 min , √ r 2 max − r 2 min r 2 − r 2 min ) , r$_{0}$ = r$_{min}$. (2.17) ϕ = ϕ$_{0}$ + Jc 2 ∫ r r 2 0 dx x √ x ( E − κ 2 2 x ) 2 − ( Jc ) 2 . (2.16) r$_{min}$ = 2 √ 2 E 3 κ 2 sin α 3 , (2.14) r$_{max}$ = 2 E 3 κ $_{$_{2}$(}$√$_{3 cos}$ α 3 − sin α 3 ) , (2.15) κ 2 2 r 3 − rE + Jc = 0 . (2.13)