The precise identification would again involve diagonalization of the dilatation operator Δ on the field theory side. Since the states are made of e $^{1}$, e3 only in SU (3), we say they are along the meridian of$_{CP}$ $^{2}$. Let us now move to the case of nontrivial potentials. For simplicity, we adjust the angular velocities so that the potential takes a relatively simple form, To achieve that one sets ζ +3 ν$_{1}$ = 0 for Case 1 and 2, and ζ = 0 for Case 3. For the problems at hand, we always find b 2 < c $^{2}$. With energy κ $^{2}$, it is convenient to introduce a parameter y = κ $^{2}$− b 2 c $^{2}$− b 2 to describe the solutions. If 0 < y < 1, the motion is vibrational, while y ≥ 1 makes it rotational. As a membrane, a vibrational motion means the worldvolume is folded and only partly covers the longitude along θ or ˜ θ . In other words, the membrane is cylindrical in total. Rotational motion would on the other hand imply a toroidal shape. We can express all conserved charges as a function of y . The energy Δ is fixed if we impose the periodicity condition. For Case 1, the motion is determined by From σ$_{2}$ ∼ σ$_{2}$ + 2 π , we have where y = 16 κ 2 ω 2 − 1 = sin 2 θ$_{0}$ . K is the complete elliptic integral of the first kind, and see the appendix for our convention and some useful formulas. Δ = √ λ $^{′}$κ can be now rewritten as One can proceed to calculate all conserved quantities for the three cases. We record the 14 V ( x ) = b $^{2}$cos2 x + c $^{2}$sin2 x. (45) κ 2 = 3 n $^{2}$e2 128 θ ′ 2 + ω 2 16 (1 + sin 2 θ ) . (46) π 2 = $^{√}$3 n $^{2}$e2 128 ∫ θ$_{0}$ 0 dθ $^{√}$κ 2 − ω 2 $_{16}$(1 + sin 2 θ ) = √ 6 ne 4 ω K ( y ) , (47) Δ √ λ = n $^{√}$6( y + 1) 8 π K ( y ) . (48) 2. Folded/wrapped rotating membranes