  A rapidly rotating Bose-Einstein condensate in a symmetric two-dimensional
harmonic trap can be described with the lowest Landau-level set of
single-particle states. The condensate wave function psi(x,y) is a Gaussian
exp(-r^2/2), multiplied by an analytic function f(z) of the complex variable z=
x+ i y. The criterion for a quantum phase transition to a non-superfluid
correlated many-body state is usually expressed in terms of the ratio of the
number of particles to the number of vortices. Here, a similar description
applies to a rapidly rotating non-symmetric two-dimensional trap with arbitrary
quadratic anisotropy (omega_x^2 < omega_y^2). The corresponding condensate wave
function psi(x,y) is a complex anisotropic Gaussian with a phase proportional
to xy, multiplied by an analytic function f(z), where z = x + i \beta_- y is a
stretched complex variable and 0< \beta_- <1 is a real parameter that depends
on the trap anisotropy and the rotation frequency. Both in the mean-field
Thomas-Fermi approximation and in the mean-field lowest Landau level
approximation with many visible vortices, an anisotropic parabolic density
profile minimizes the energy. An elongated condensate grows along the soft trap
direction yet ultimately shrinks along the tight trap direction. The criterion
for the quantum phase transition to a correlated state is generalized (1) in
terms of N/L_z, which suggests that a non-symmetric trap should make it easier
to observe this transition or (2) in terms of a "fragmented" correlated state,
which suggests that a non-symmetric trap should make it harder to observe this
transition. An alternative scenario involves a crossover to a quasi
one-dimensional condensate without visible vortices, as suggested by Aftalion
et al., Phys. Rev. A 79, 011603(R) (2009).
