  We report on a study of a bilayer two-dimensional electron gas at Landau
level filling factor $\nu=2$. The system exhibits both magnetic and spontaneous
interlayer phase coherence broken symmetries. We propose a 3-parameter Slater
determinant variational wavefunction which describes the ground state over the
full range of bias potential ($\Delta_V$) Zeeman coupling ($\Delta_z$) and
interlayer tunneling amplitude ($\Delta_t$) strengths. Broken symmetry states
occur inside a volume in this three-dimensional phase diagram near the
$\Delta_z^2=\Delta_V^2+\Delta_t^2$ surface. We have obtained analytic results
for the intersections of the phase diagram with the $\Delta_t=0$, $\Delta_z=0$,
and $\Delta_V=0$, planes and show that the differential capacitance of the
bilayer system is singular at the phase boundary.
