  We investigate the energy spectrum of a single and two electron quantum dot
(QD) embedded in two dimensional electron gas at the interface between
SrTiO$_3$ and LaAlO$_3$, in the presence of the external magnetic field. For
this purpose the three band model of $3d$-electrons defined on the square
lattice of Ti ions was utilized. We demonstrate that, for the weak parabolic
confinement potential, the low energy spectrum is sufficiently well described
by the effective Hamiltonian reduced to the one $d_{xy}$ orbital with the
spin-orbit interaction originating from the coupling to the $d_{xz}$, $d_{yz}$
bands. This is not the case for stronger confinement where contribution of the
states related to the $d_{xz/yz}$ orbital is relevant. Based on the time
depended calculations we discuss in details the manipulation of the electron
spin in QD by external AC voltages, in the context of the electric dipole spin
resonance. The allowed and forbidden transitions are discussed in details with
respect to the parity selection rule. Our calculations show that for a single
electron QD the spin-flip in the ground-state has a character of a Rabi
resonance while for two electrons the singlet-triplet transition is forbidden
by the parity symmetry. For the two electrons QD, we demonstrate that the
spin-flip transition can still be accomplished via a second-order, two-photon
process that has a two-state Rabi character for low AC field amplitude. The
violation of the parity symmetry on the spin-flip transitions is also analyzed.
