  Weakly nonlinear analysis of resonant PDEs in recent literature has generated
a number of resonant systems for slow evolution of the normal mode amplitudes
that possess remarkable properties. Despite being infinite-dimensional
Hamiltonian systems with cubic nonlinearities in the equations of motion, these
resonant systems admit special analytic solutions, which furthermore display
periodic perfect energy returns to the initial configurations. Here, we
construct a very large class of resonant systems that shares these properties
that have so far been seen in specific examples emerging from a few standard
equations of mathematical physics (the Gross-Pitaevskii equation, nonlinear
wave equations in Anti-de Sitter spacetime). Our analysis provides an
additional conserved quantity for all of these systems, which has been
previously known for the resonant system of the two-dimensional
Gross-Pitaevskii equation, but not for any other cases.
