  We investigate the ground-state structure of the bilayer quantum Hall system
at the filling factor $\nu =2$. Making an exact analysis of the ground state in
the SU(4)-invariant limit, we include all other interactions as small
perturbation. We carry out analytic calculations and construct phase diagrams
for nonzero values of the Zeeman, tunneling and bias interactions. In
particular we examine carefully how the phase transition occurs by applying the
bias voltage and inducing a density imbalance between the two layers. We
compare our theoretical result with the experimental data due to Sawada et al.
based on the phase diagram in the $\sigma_{0}$-$\rho_{0}$ plane, where $\rho
_{0}$ and $\sigma_{0}$ are the total electron density and the density
difference between the two layers, respectively.
