with the BPS relation R = Δ , Q$_{3}$ = − 3Δ 4 , Q$_{8}$ = √ 3Δ 4 , J$_{3}$ = Δ $_{2}$. Finally, when we consider membranes located at the south pole of$_{CP}$ $^{2}$, we can match and check R = Δ , Q$_{3}$ = 0 , Q$_{8}$ = −$^{√}$ 3Δ 2 , J$_{3}$ = Δ $_{2}$. For all the three examples given above, we can switch to lowest weight states of SU (2), by choosing θ = π . Then one needs to change J$_{3}$ = − Δ $_{2}$, and use e $^{ia}$X i $_{12}$X i $_{31}$X i $_{31}$, i = 1 , 2 , 3 for the field theory identification. We now turn to the case where membranes occupy a genuinely three-dimensional world- volume. One first notes that in full generality the membrane action Eq. (19) is still not easy to deal with. It is because of the determinant part det G$_{ij}$ , which gives rise to terms with four derivatives in the action. For example det G$_{ij}$ contains terms like L 4 64 [( ∂$_{σ}$$_{1}$θ ) $^{2}$( ∂$_{σ}$$_{2}$θ ) 2 − ( ∂$_{σ}$$_{1}$∂$_{σ}$$_{2}$θ ) $^{2}$], and it is obvious that even without the constraints Eq. (20- 21) the equations of motion are quite nontrivial to handle. Our strategy to discover nontrivial rotating solutions is as follows. First, we adopt the temporal gauge and assume linear time dependence for ψ, ˜ ψ, φ, ˜ φ as in Eq. (23). Then the constraint Eq. (21) is trivially satisfied, while Eq. (20) is independent of τ . Secondly, in order to avoid solving nonlinear partial differential equations, we assume each of µ, θ, ˜ θ is either constant, a linear function of σ$_{1}$ , or an undetermined function of σ$_{2}$ only. If we also set some of the angular velocities ˙ ψ, ˙ ˜ ψ, ˙ φ, ˙ ˜ φ to zero, it is possible to make the constraint Eq. (20) independent of σ$_{1}$ , and a function of σ$_{2}$ only. Then we check if the equations from action Eq. (19) is compatible with Eq. (20). More concretely, in this prescription the equations of motion can be treated as classical mechanics system with time variable σ$_{2}$ . Consistency of the ansatz implies Eq. (20) corresponds to the energy integral. Our survey has resulted in three physically distinct cases where one can consistently re- duce the membrane equations of motion into that of an auxiliary one-dimensional mechanical system. All of them are in general extended both in CP 2 and CP $^{1}$, and have multiple non- vanishing angular momenta. We provide the concrete ansatze for the angles in the below. We can always give non-vanishing angular velocity for ψ = ζτ . For other azimuthal angles like ˜ ψ, ˜ φ, φ , it is needed to set some of their velocities to zero, depending on the ansatz. We 10 B. Membranes extended and rotating in M 1 , 1 , 1 µ = 0 , θ = 0 ⇒ Tr $^{[}$( e $^{ia}$X 3 $_{12}$X 3 $_{23}$X 3 $_{23}$) n $^{]}$, (35)