TABLE I. Rotating membranes as a pendulum Type Variable Mass a b c Case 1 θ 3 n $^{2}$e2 64 ζ +3 ν$_{1}$ 8 ω 4 √ 2 ω 4 Case 2 ˜ θ 3 n $^{2}$e2 64 ζ +3 ν$_{1}$ 8 3 ν$_{2}$ 8 √ 3 ν$_{2}$ 4 Case 3 θ 3 n $^{2}$e2 256 ζ 8 ω 4 √ 2 ω 4 the same mathematical techniques here. Although the computation is in principle straight- forward, the results are rather messy for the most general case of Eq. (38). In this article we will only consider two special subclasses: (i) b = c = 0 and (ii) a = 0. Full analysis with Eq. (38) and extension to more general solutions analogous to giant magnons [22] will be reported in a separate publication. If we set ω = 0 in Eq. (37) the potential becomes a constant and we have a particularly simple solution, θ = mσ$_{2}$ with m ∈$_{Z}$ . The same solutions are obtained if we set ν$_{2}$ = 0 in Case 2. Similar but physically distinct solutions are given from Case 3 if we set ω = 0. Both of these solutions uniformly wrap two polar angles, and we will call them toroidal rotating membranes . Below we will consider such single-spin membranes and provide the dual gauge theory interpretation. After that, we will consider cases where a = 0. For this class of solutions we have ν$_{2}$ = ω = 0. And without losing generality we can further set ν$_{1}$ = 0. Among the seven coordinates of M 1 , 1 , $^{1}$, we see that nontrivial ones are ψ = ζτ, ˜ θ = nσ$_{1}$, θ = mσ$_{2}$ , and µ = π $_{2}$. One can readily establish the following relationships between conserved charges, where λ = ( L $^{3}$T$_{2}$ ) $^{2}$, the ’t Hooft coupling constant. Other angular momenta are given by 12 1. Toroidal rotating membranes • On the equator of$_{CP}$ $^{2}$: ( Δ √ λ$^{)}$$^{2}$ =$^{(}$ R √ λ$^{)}$$^{2}$ + 3 n $^{2}$m2 128 , (39) R : Q$_{3}$ : Q$_{8}$ : J$_{3}$ = 1 : 0 : 3 4 R : 0 . (40)