  Numerical simulations demonstrate a link between dynamically cold initial
solutions and self-similarity. However the nature of this link is not fully
understood. Cold initial conditions alone without further symmetry do not lead
to self-similarity. Here we show that when the system approaches equilibrium a
new symmetry appears. The combination of this equilibrium symmetry with the
cold symmetry in the initial conditions leads to full self-similarity. As a
consequence for any initially cold system even if the initial spatial
distribution is not self-similar we will observe an evolution towards
self-similarity near equilibrium. The case of one dimensional systems or
spherically symmetric systems in 3D are discussed in detail. Systems depending
on the energy and other integrals are also considered. The problem of the
degeneracy of the self-similar solutions at equilibrium is tackled. It is shown
that very small perturbations at the center of the system have the ability to
break this degeneracy and lead to the convergence towards a specific
auto-similar solution.
