  We consider the cubic Gross-Pitaevskii (GP) hierarchy in one spatial
dimension. We establish the existence of an infinite sequence of observables
such that the corresponding trace functionals, which we call ``energies,''
commute with respect to the weak Lie-Poisson structure defined by the authors
in arXiv:1908.03847. The Hamiltonian equation associated to the third energy
functional is precisely the GP hierarchy. The equations of motion corresponding
to the remaining energies generalize the well-known nonlinear Schr\"odinger
hierarchy, the third element of which is the one-dimensional cubic nonlinear
Schr\"odinger equation. This work provides substantial evidence for the GP
hierarchy as a new integrable system.
