  A general approach is proposed to solve the coupled Gross-Pitaevskii
equations (CGP) for K-species Bose-Einstein condensates (BEC). Analytical
solutions have been obtained under the Thomas-Fermi approximation. We aim at
finding out the common features of the K-species BEC. In particular, two types
of phase-transitions, full-state-transition and partial-state transition, are
found. In the former all species are involved in the transition, while in the
latter only a few specified species are essentially involved. This leads to the
criticality and the hidden criticality (previously found in multi-band
superconductivity). We further found that the former originates from the
singularity of the whole matrix of the CGP, while the latter originates from
the singularity of a specified sub-matrix (which is contributed by only a few
specified species). It is emphasized that the singularity is not a by-product
of the TFA, but is inherent in the CGP.
