  Q-balls formed from the Affleck-Dine field have rich cosmological
implications and have been extensively studied from both theoretical and
simulational approaches. From the theoretical point of view, the exact solution
of the Q-ball was obtained and it shows a circular orbit in the complex plane
of the field value. In practice, however, it is reported that the Q-ball that
appears after the Affleck-Dine mechanism has an $elliptical$ orbit, which
carries larger energy per unit $U(1)$ charge than the well-known solution with
a circular orbit. We call them "elliptical" Q-balls. In this paper, we report
the first detailed investigation of the properties of the elliptical Q-balls by
$3$D lattice simulation. The simulation results indicate that the elliptical
Q-ball has an almost spherical spatial profile with no nodes, and we observed a
highly elliptic orbit that cannot be described through small perturbations
around the ground state Q-ball. Higher ellipticity leads to more excitation of
the energy, whose relation is also derived as a dispersion relation. Finally,
we derive two types of approximate solutions by extending the Gaussian
approximation and considering the time-averaged equation of motion and we also
show the consistency with the simulation results.
