Now, the four-dimensional system (2.2), where x = ( x $^{1}$, x$^{2}$), and p = ( p $^{1}$, p$^{2}$) is completely integrable. Indeed, it possesses two integrals in involution E and J . Therefore, the motion between two circles with the radius r$_{min}$ and r$_{max}$ can be only quasiperiodic and periodic. Of course the case of the periodic motion refers to a closed path. This means that an angle Δ ϕ given by (see formula (2.17)) should be a rational function of π , i.e. Δ ϕ = πm/n , where m and n are integers. An example of a periodic motion between two circles is presented in Figs 1 and 2. It should be noted that the length of momentum (kinetic energy of the particle c | p | ) has maximum at r = r$_{min}$ , decreases (increases) as r approaches r$_{max}$ ( r$_{min}$ ), and for r = r$_{max}$ has minimum. Clearly, such behavior of the momentum of a massless particle is consistent with the form of (2.6). The values of r$_{min}$ and r$_{max}$ as well as extrema of the length of momentum can be expressed as a function of the energy by means of the implicit formulas (2.14) and (2.15). We finally remark that the case of the uniform motion in a circle discussed earlier [type 1) of the motion] refers to the condition r$_{min}$ = r$_{max}$ = R = √ 2 E 3 κ 2 . The third type of the motion corresponds to J = 0, so we have 3) r ( E − κ 2 2 r $^{2}$) ≥ 0. From this inequality we find 0 ≤ r ≤ r$_{max}$ = √$_{2}$$_{E}$ κ . On the other hand, taking into account (2.7) we find that for J = 0 the system (2.5) reduces to where θ − ϕ = 0 or | θ − ϕ | = π . Therefore a particle motion is uniform in a segment [0 , √$_{2}$$_{E}$ κ ], more precisely, we have r = ± ct + r$_{0}$ , where 0 ≤ r ≤ √$_{2}$$_{E}$ κ and the two signs correspond to two possible directions of motion. Assuming that a particle moves in the x -coordinate line, ˙ r = ± c, ˙ ϕ = 0 , ˙ p = ∓ κ $^{2}$r, ˙ θ = 0 , (2.21) 6 Δ ϕ = Jc 2 ∫ r 2 max r 2 min dx x √ x ( E − κ 2 2 x ) 2 − ( Jc ) 2 = 2 Jc κ $^{2}$r 2 min √ r 2 − − r 2 min Π ( π 2 , 1 − r 2 max r 2 min , √ r 2 max − r 2 min r 2 − r 2 min ) , (2.20)