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For the spinor Bose-Einstein condensates both the total spin $S$ and itsZ-component $S_{Z}$ should be conserved. However, in existing theories, onlythe conservation of $S_{z}$ has been taken into account. To remedy, this paperis the first attempt to take the conservation of both $% S$ and $S_{Z}$ intoaccount. 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 generalizedGross-Pitaevskii equation has been derived. With this new equation, the groundbands of the $^{23}$Na and $% ^{87}$Rb condensates have been studied, where thelevels distinct in $S$ split. It was found that the level density is extremelydense in the bottom of the ground band of $^{23}$Na, i.e., in the vicinity ofthe ground state. On the contrary, for $^{87}$Rb, the levels are extremelydense in the top of the ground band,Comment: 7 page, 5 figure

Ground band and a generalized Gross-Pitaevskii equation for spinor Bose-Einstein condensates