  A generalized Gross-Pitaevskii equation adapted to the $U(5)\supset
SO(5)\supset SO(3)$ symmetry has been derived and solved for the spin-2
condensates. 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 gap
and the level density in the neighborhood of the g.s., have been studied. Based
on a rigorous treatment of the spin-degrees of freedom, the spin-textiles can
be understood in a $N$-body language. In addition to the ferro-, polar, and
cyclic phases, the g,s, might in a mixture of them when $0< M< 2N$ ($M$ is the
total magnetization). The great difference in the stability and degeneracy of
the g.s. caused by varying $\varphi $ (which marks the features of the
interaction) and $M$ is notable.
  Since the root mean square radius $R_{rms}$ is an observable, efforts have
been 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 check
the theories with experimental data.
