  We demonstrate that dynamical symmetry plays a crucial role in determining
the structure of the eigenspectra of spinor Bose-Einstein condensates (BECs).
In particular, the eigenspectra of spin- 1 and spin-2 BECs in the single-mode
approximation are shown to be completely determined by dynamical symmetries,
where a spin-2 BEC corresponds to the U(5) limit of the interacting boson model
in nuclear physics. The eigenspectrum of a spin-3 BEC is solved analytically
for a specific class of coupling constants, while it is shown that dynamical
symmetry alone is not enough to determine the spectrum for arbitrary coupling
constants. We also study the low-lying eigenspectra of spin-1 and spin-2 BECs
in the absence of external magnetic field, and find, in particular, that the
quasidegenerate spectra emerge for antiferromagnetic and cyclic phases. This
implies that these systems are highly susceptible to external perturbations and
may undergo symmetry-breaking transitions to other states upon increasing the
size of system.
