  We study optical conductivity of a two-dimensional electron gas with
anisotropic $k$-cubic Rashba spin-orbit interaction formed at the
LaAlO$_3$/SrTiO$_3$ interface. The anisotropic spin splitting energy gives rise
to different features of the optical conductivity in comparison to the
isotropic $k$-cubic Rashba spin-orbit interaction. For large carrier density
and strong spin-orbit couplings, the density dependence of Drude weight
deviates from the linear behavior. The charge and optical conductivities remain
isotropic despite anisotropic nature of the Fermi contours. An infinitesimally
small photon energy would suffice to initiate inter-band optical transitions
due to degeneracy along certain directions in momentum space. The optical
conductivity shows a single peak at a given photon energy depending on the
system parameters and then falls off to zero at higher photon energy. These
features are lacking for systems with isotropic $k$-cubic Rashba spin-orbit
coupling. These striking features can be used to extract the information about
nature of the spin-orbit interaction experimentally and illuminate some light
on the orbital origin of the two-dimensional electron gas.
