  We study the continuous resonant (CR) equation which was derived by
Faou-Germain-Hani as the large-box limit of the cubic nonlinear Schr\"odinger
equation in the small nonlinearity (or small data) regime. We first show that
the system arises in another natural way, as it also corresponds to the
resonant cubic Hermite-Schr\"odinger equation (NLS with harmonic trapping). We
then establish that the basis of special Hermite functions is well suited to
its analysis, and uncover more of the striking structure of the equation. We
study in particular the dynamics on a few invariant subspaces: eigenspaces of
the harmonic oscillator, of the rotation operator, and the Bargmann-Fock space.
We focus on stationary waves and their stability.
