  We consider a class of distribution functions having the form $
f(v,t)=t^{-dat} F(v e^{-at} )$, $ a=\mathrm{const.} $, where $ v $ and $ t $
denote the particle velocity and the time. This class of self-similar solutions
to the spatially homogeneous Boltzmann equation (BE) for Maxwell molecules was
studied by Bobylev and Cercignani in early 2000s. The solutions are positive,
but have an infinite second moment (energy). However, the same class of
distribution functions with finite energy appears to be closely connected with
quite different class of group-invariant solutions of the spatially
inhomogeneous BE. This is a motivation for considering the so-called modified
spatially homogeneous BE, which contains an extra force term proportional to a
matrix $ A $. The modified BE was recently studied under assumption of
"sufficient smallness of norm $ \|A\| $" without explicit estimates of the
smallness. We fill this gap and prove that all important facts related to
self-similar solutions remain valid also for moderate values of $ \|A\| $.
