  For the spinor Bose-Einstein condensates both the total spin $S$ and its
Z-component $S_{Z}$ should be conserved. However, in existing theories, only
the conservation of $S_{z}$ has been taken into account. To remedy, this paper
is the first attempt to take the conservation of both $% S$ and $S_{Z}$ into
account. For this purpose, a total spin-state with the good quantum numbers $S$
and $S_{Z}$ is introduced in the trial wave function, thereby a generalized
Gross-Pitaevskii equation has been derived. With this new equation, the ground
bands of the $^{23}$Na and $% ^{87}$Rb condensates have been studied, where the
levels distinct in $S$ split. It was found that the level density is extremely
dense in the bottom of the ground band of $^{23}$Na, i.e., in the vicinity of
the ground state. On the contrary, for $^{87}$Rb, the levels are extremely
dense in the top of the ground band,
