  We consider the singular Rayleigh-Jeans equilibrium of the $4$-waves kinetic
turbulence equation for the three dimensional Schr\"{o}dinger equation. We
first show the formation in finite time of a Dirac measure at zero frequency in
the solution of the wave kinetic equation when the initial data has the form of
Rayleigh-Jeans, truncated at large values of the energy. The initial value
problem for the linearization around the singular Rayleigh-Jeans equilibria is
then solved in several functional spaces. Then, long time convergence to a
Dirac measure at the origin is described in detail for some of the solutions.
This determines a basin of attraction of the Dirac measure.
