third and more realistic model we include the perturbations of eight planets, the Moon, (1) Ceres, (2) Pallas and (4) Vesta on the asteroid for the nominal orbit in Table \[elements\]. Figure \[mld3\] shows that the evolution of the mean longitude difference in the time interval -7,000 to +1,000 yr is very similar to those from Models 1 and 2 (see Figs. \[mld1\] and \[mld2\]) confirming again that 2002 VE$_{68}$ has been a quasi-satellite of Venus for a period of time but that it will soon leave the quasi-satellite phase for the L$_5$ Lagrangian point. The behavior of the mean longitude difference in that time range is consistent across models and differences only appear outside that time interval. The mean longitude of the asteroid currently librates around the value of the mean longitude of Venus with an amplitude of 40$^{\circ}$-60$^{\circ}$ and an average period of about 150 yr.

The orbital evolution of the asteroid after leaving the quasi-satellite phase is significantly more complex than in previous models with multiple transitions between the various co-orbital resonant states. The evolution of the orbital elements in Fig. \[orbele3\] is quite similar to that in Fig. \[orbele2\] for Model 2. Figure \[distance3\] is consistent 