  We consider the nonlinear Schr\"{o}dinger equation of degree five on the
circle $\mathbb{S}^1 = \mathbb{R}/2\pi$. We prove the existence of
quasi-periodic solutions which bifurcate from "resonant" solutions (studied in
[14]) of the system obtained by truncating the Hamiltonian after one step of
Birkhoff normal form, exhibiting recurrent exchange of energy between some
Fourier modes. The existence of these quasi-periodic solutions is a purely
nonlinear effect.
