Missing top of the AdS resonance structure

We study a massless scalar field in AdS(d+1) with a nonlinear coupling phi(N) and not limited to spherical symmetry. The free-field-eigenstate spectrum is strongly resonant, and it is commonly believed that the nonlinear coupling leads to energy transfer between eigenstates. We prove that when Nd is even, the most efficient resonant channels to transfer energy are always absent. In particular, for N = 3 this means no energy transfer at all. For N = 4, this effectively kills half of the channels, leading to the same set of extra conservation laws recently derived for gravitational interactions within spherical symmetry.