  We study two resonant Hamiltonian systems on the phase space $L^2(\mathbb{R}
\rightarrow \mathbb{C})$: the quintic one-dimensional continuous resonant
equation, and a cubic resonant system that has appeared in the literature as a
modified scattering limit for an NLS equation with cigar shaped trap. We prove
that these systems approximate the dynamics of the quintic and cubic
one-dimensional NLS with harmonic trapping in the small data regime on long
times scales. We then pursue a thorough study of the dynamics of the resonant
systems themselves. Our central finding is that these resonant equations fit
into a larger class of Hamiltonian systems that have many striking dynamical
features: non-trivial symmetries such as invariance under the Fourier transform
and the flow of the linear Scr\"odinger equation with harmonic trapping, a
robust wellposedness theory, including global wellposedness in $L^2$ and all
higher $L^2$ Sobolev spaces, and an infinite family of orthogonal, explicit
stationary wave solutions in the form of the Hermite functions.
