  We find the analytical solution to the time-dependent density-functional
theory (TDDFT) problem for the quasi-low-dimensional (2D and 1D) electron gas
(QLDEG) with only one band filled in the direction perpendicular to the system
extent. The theory is developed at the level of TD exact exchange and yields
the exchange potential as an explicit nonlocal operator of the spin-density.
The dressed interband (image states) excitation spectra of the Q2DEG are
calculated, while the comparison with the Kohn-Sham (KS) transitions provides
the insight into the qualitative and quantitative role of the many-body
interactions. Important cancellations between the Hartree $f_H$ and the
exchange $f_x$ kernels are found in the low-density limit, shedding light on
the interrelations between the KS and many-body excitations.
