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A generalized Gross-Pitaevskii equation adapted to the $U(5)\supsetSO(5)\supset SO(3)$ symmetry has been derived and solved for the spin-2condensates. The spin-textile and the degeneracy of the ground state (g.s.)together with the factors affecting the stability of the g.s., such as the gapand the level density in the neighborhood of the g.s., have been studied. Basedon a rigorous treatment of the spin-degrees of freedom, the spin-textiles canbe understood in a $N$-body language. In addition to the ferro-, polar, andcyclic phases, the g,s, might in a mixture of them when $0< M< 2N$ ($M$ is thetotal magnetization). The great difference in the stability and degeneracy ofthe g.s. caused by varying $\varphi $ (which marks the features of theinteraction) and $M$ is notable.  Since the root mean square radius $R_{rms}$ is an observable, efforts havebeen made to derive a set of formulae to relate $R_{rms}$ and $% N$, $\omega$(frequency of the trap), and $\varphi $. These formulae provide a way to checkthe theories with experimental data.

Generalized Gross-Pitaevskii equation adapted to the $U(5)\supset SO(5)\supset SO(3)$ symmetry for spin-2 condensates