  This is a sequel to the paper "Large time asymptotics for a cubic nonlinear
Schr\"odinger system in one space dimension" by the same authors. We continue
to study the Cauchy problem for the two-component system of cubic nonlinear
Schr\"odinger equations in one space dimension. We provide criteria for large
time decay or non-decay in $L^2$ of the small amplitude solutions in terms of
the Fourier transforms of the initial data.
