  Multilayer graphene lattices allow for an additional tunability of the band
structure by the strong perpendicular electric field. In particular, the
emergence of the new multiple Dirac points in ABA stacked trilayer graphene
subject to strong transverse electric fields was proposed theoretically and
confirmed experimentally. These new Dirac points dubbed ``gullies'' emerge from
the interplay between strong electric field and trigonal warping. In this work
we first characterize the properties of new emergent Dirac points and show that
the electric field can be used to tune the distance between gullies in the
momentum space. We demonstrate that the band structure has multiple Lifshitz
transitions and higher-order singularity of ``monkey saddle'' type. Following
the characterization of the band structure, we consider the spectrum of Landau
levels and structure of their wave functions. In the limit of strong electric
fields when gullies are well separated in momentum space, they give rise to
triply degenerate Landau levels. In the second part of this work, we
investigate how degeneracy between three gully Landau levels is lifted in
presence of interactions. Within the Hartree-Fock approximation we show that
the symmetry breaking state interpolates between fully gully polarized state
that breaks $C_3$ symmetry at high displacement field, and the gully symmetric
state when the electric field is decreased. The discontinuous transition
between these two states is driven by enhanced inter-gully tunneling and
exchange. We conclude by outlining specific experimental predictions for the
existence of such a symmetry-breaking state.
