  Initially cold and spherically symmetric self-gravitating systems may give
rise to a virial equilibrium state which is far from spherically symmetric, and
typically triaxial. We focus here on how the degree of symmetry breaking in the
final state depends on the initial density profile. We note that the most
asymmetric structures result when, during the collapse phase, there is a strong
injection of energy preferentially into the particles which are localized
initially in the outer shells. These particles are still collapsing when the
others, initially located in the inner part, are already re-expanding; the
motion of particles in a time varying potential allow them to gain kinetic
energy --- in some cases enough to be ejected from the system. We show that
this mechanism of energy gain amplifies the initial small deviations from
perfect spherical symmetry due to finite $N$ fluctuations. This amplification
is more efficient when the initial density profile depends on radius, because
particles have a greater spread of fall times compared to a uniform density
profile, for which very close to symmetric final states are obtained}. These
effects lead to a distinctive correlation of the orientation of the final
structure with the distribution of ejected mass, and also with the initial
(very small) angular fluctuations.
