  In this paper, we have studied the integrability nature of a system of three
coupled Gross-Pitaevskii type nonlinear evolution equations arising in the
context of spinor Bose-Einstein condensates by applying the Painlev\'e
singularity structure analysis. We show that only for two sets of parametric
choices, corresponding to the known integrable cases, the system passes the
Painlev\'e test.
