  The two-dimensional electron gas in a bilayer graphene in the Bernal stacking
supports a variety of uniform broken-symmetry ground states in Landau level N=0
at integer filling factors $\nu \in [-3,4].$ When an electric potential
difference (or bias) is applied between the layers at filling factors $\nu
=-1,3$, the ground state evolves from an interlayer coherent state at small
bias to a state with orbital coherence at higher bias where \textit{electric}
dipoles associated with the orbital pseudospins order spontaneously in the
plane of the layers. In this paper, we show that by further increasing the bias
at these two filling factors, the two-dimensional electron gas goes first
through a Skyrmion crystal state and then into an helical state where the
pseudospins rotate in space. The pseudospin textures in both the Skyrmion and
helical states are due to the presence of a Dzyaloshinskii-Moriya interaction
in the effective pseudospin Hamiltonian when orbital coherence is present in
the ground state. We study in detail the electronic structure of the helical
and Skyrmion crystal states as well as their collective excitations and then
compute their electromagnetic absorption.
