Missing top of the AdS resonance structure

We study a massless scalar field in AdSd+1 with a nonlinear coupling φN 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.