  The model considered in the paper is used nowadays to describe spin dynamics
of quantum dots after optical excitation. Based on the exact diagonalization of
a model Hamiltonian, we solve the problems of the electron spin polarization
decay and magnetic field dependence of the steady state polarization. The
important role of the nuclear state is shown and methods of its calculation for
different regimes of optical excitation are proposed. The effect of spin echo
observed after application of the magnetic field $\pi$-pulse is predicted.
