  We construct a family of self-similar solutions with fat tails to a quadratic
kinetic equation. This equation describes the long time behaviour of weak
solutions with finite mass to the weak turbulence equation associated to the
nonlinear Schr\"odinger equation. The solutions that we construct have finite
mass, but infinite energy. In J. Stat. Phys. 159:668-712, self-similar
solutions with finite mass and energy were constructed. Here we prove upper and
lower exponential bounds on the tails of these solutions.
