  In this paper, we rigorously obtain all the equilibria of collision kernels
of type "two particles give two particles" appearing in weak turbulence theory
under very general assumptions, thus completing the "equality case" in
Boltzmann's H-theorem for those models. We also provide some rigorous results
for collision kernels of type "two particles give one particle", under
assumptions which include some of the most classical kernels of this type. The
method of proof is inspired by the quantitative estimates obtained for the
Landau equation by Desvillettes in [J. Funct. Anal. 269:1359-1403, 2015].
