  In a quantizing magnetic field, the chiral two-dimensional electron gas in
Landau level $N=0$ of bilayer graphene goes through a series of phase
transitions at integer filling factors $\nu \in \left[ -3,3\right] $ when the
strength of an electric field applied perpendicularly to the layers is
increased. At filling factor $\nu =3,$ the electron gas can described by a
simple two-level system where layer and spin degrees of freedom are frozen. The
gas then behaves as an orbital quantum Hall ferromagnet. A Coulomb-induced
Dzyaloshinskii-Moriya term in the orbital pseudospin Hamiltonian is responsible
for a series of transitions first to a Wigner crystal state and then to a
spiral state as the electric field is increased. Both states have a non trivial
orbital pseudospin texture. In this work, we study how the phase diagram at
$\nu =3$ is modified by an electric field applied in the plane of the layers
and then derive several experimental signatures of the uniform and nonuniform
states in the phase diagram. In addition to the transport gap, we study the
electromagnetic absorption and the Kerr rotation due to the excitations of the
orbital pseudospin-wave modes in the broken-symmetry states.
