  In this paper, we discuss the results of a new particle pusher in realistic
ultra-strong electromagnetic fields as those encountered around rotating
neutron stars. After presenting results of this algorithm in simple fields and
comparing them to expected exact analytical solutions, we present new
simulations for a rotating magnetic dipole in vacuum for a millisecond pulsar
by using Deutsch solution. Particles are injected within the magnetosphere,
neglecting radiation reaction, interaction among them and their feedback on the
fields. Our simulations are therefore not yet fully self-consistent because
Maxwell equations are not solved according to the current produced by these
particles. The code highlights the symmetrical behaviour of particles of
opposite charge to mass ratio $q/m$ with respect to the north and south
hemispheres. The relativistic Lorentz factor of the accelerated particles is
proportional to this ratio $q/m$: protons reach up to $\gamma_p \simeq
10^{10.7}$, whereas electrons reach up to $\gamma_e \simeq 10^{14}$.
  Our simulations show that particles could be either captured by the neutron
star, trapped around it, or ejected far from it, well outside the
light-cylinder. Actually, for a given charge to mass ratio, particles follow
similar trajectories. These particle orbits show some depleted directions,
especially at high magnetic inclination with respect to the rotation axis for
positive charges and at low inclination for negative charges because of
symmetry. Other directions are preferred and loaded with a high density of
particles, some directions concentrating the highest or lowest acceleration
efficiencies.
