  We investigate the effect of the anisotropy of a harmonic trap on the
behaviour of a fast rotating Bose-Einstein condensate. This is done in the
framework of the 2D Gross-Pitaevskii equation and requires a symplectic
reduction of the quadratic form defining the energy. This reduction allows us
to simplify the energy on a Bargmann space and study the asymptotics of large
rotational velocity. We characterize two regimes of velocity and anisotropy; in
the first one where the behaviour is similar to the isotropic case, we
construct an upper bound: a hexagonal Abrikosov lattice of vortices, with an
inverted parabola profile. The second regime deals with very large velocities,
a case in which we prove that the ground state does not display vortices in the
bulk, with a 1D limiting problem. In that case, we show that the coarse grained
atomic density behaves like an inverted parabola with large radius in the
deconfined direction but keeps a fixed profile given by a Gaussian in the other
direction. The features of this second regime appear as new phenomena.
