  We investigate a two-dimensional system of with two values of the internal
(spin) degree of freedom. It is confined by a deformed harmonic trap and
subject to a Zeeman field, Rashba or Dresselhaus one-body spin-orbit couplings
and two-body short range repulsion. We obtain self-consistent mean-field
$N$-body solutions as functions of the interaction parameters. Single-particle
Spectra and total energies are computed and compared to the results without
interaction. We perform a statistical analysis for the distributions of nearest
neighbor energy level spacings and show that quantum signatures of chaos are
seen in certain parameters regimes. Furthermore, the effects of two-body
repulsion on the nearest neighbor distributions are investigated. This
repulsion can either promote or destroy the signatures of potential chaotic
behavior depending on relative strengths of parameters. Our findings support
the suggestion that cold atoms may be used to study quantum chaos both in the
presence and absence of interactions.
