  We consider the dynamics of a spherically symmetric massless scalar field
coupled to general relativity in Anti--de Sitter spacetime in the
small-amplitude limit. Within the context of our previously developed two time
framework (TTF) to study the leading self-gravitating effects, we demonstrate
the existence of two new conserved quantities in addition to the known total
energy $E$ of the modes: The particle number $N$ and Hamiltonian $H$ of our TTF
system. Simultaneous conservation of $E$ and $N$ implies that weak turbulent
processes undergo dual cascades (direct cascade of $E$ and inverse cascade of
$N$ or vice versa). This partially explains the observed dynamics of 2-mode
initial data. In addition, conservation of $E$ and $N$ limits the region of
phase space that can be explored within the TTF approximation and in particular
rules out equipartion of energy among the modes for general initial data.
Finally, we discuss possible effects of conservation of $N$ and $E$ on late
time dynamics.
