  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.
