  We study rotational motion of an interacting atomic Bose-Einstein condensate
confined in a quadratic-plus-quartic potential. We calculate the lowest energy
surface mode frequency and show that a symmetric trapped (harmonic and quartic)
Bose-Einstein condensate breaks the rotational symmetry of the Hamiltonian when
rotational frequency is greater than one-half of the lowest energy surface mode
frequency. We argue that the formation of a vortex is not possible in a
non-interacting as well as in an attractive Bose-Einstein condensate confined
in a harmonic trap due to the absence of the spontaneous shape deformation, but
it can occur which leads to the vortex formation if we add an additional
quartic potential. Moreover, the spontaneous shape deformation and consequently
the formation of a vortex in an attractive system depends on the strengths of
the two-body interaction and the quartic potential.
