  We analyze the velocity distribution function of force-free granular gases in
the regime of homogeneous cooling when deviations from the Maxwellian
distribution may be accounted only by leading term in the Sonine polynomial
expansion. These are quantified by the magnitude of the coefficient $a_2$ of
the second term of the expansion. In our study we go beyond the linear
approximation for $a_2$ and observe that there are three different values
(three roots) for this coefficient which correspond to a scaling solution to
the Boltzmann equation. The stability analysis performed showed, however, that
among these three roots only one corresponds to a stable scaling solution. This
is very close to $a_2$, obtained in previous studies in a linear with respect
to $a_2$ approximation.
