We can now obtain an alternative form of Polyakov-type action if we perform the Legendre transformation once more, Varying this action with respect to λ $^{k}$, we obtain the constraints: We can work in the gauge, λ i = 0 , λ 0 = const [15]. Then G$_{0}$$_{i}$ =0 and the worldvolume metric takes a block-diagonal form, and the action is simplified as This is the form of the gauge-fixed membrane action we will use in this paper. The equations of motion derived from Eq. (19) should be compatible with the the following constraints, Let us now present the ansatz we will use for the spinning membrane solutions. In the action Eq. (19), the coordinates of the 11 dimensional spacetime are treated as fields on three dimensional worldvolume coordinates τ, σ$_{1}$, σ$_{2}$ . We will be using the static gauge and identify as t = κτ . The motion is in M 1 , 1 , 1 only, so we will set other coordinates of AdS$_{4}$ to constants, i.e. ρ = ϑ = ϕ = 0. This setup is consistent with the action and constraints, and we effectively have membrane motion in$_{R}$ × M 1 , 1 , $^{1}$, with metric (apart from the scale factor L ) To describe rotations, we will give linear τ -dependence for azimuthal angles, and set 7 IV. MEMBRANES ROTATING IN M 1 , 1 , 1 AND THEIR ANGULAR MOMENTA S B = ∫ d $^{3}$σ ( P$_{M}$ ∂$_{0}$X M − H ) = ∫ d $^{3}$σ 4 λ 0 $^{[}$G$_{00}$ − 2 λ $^{j}$G$_{0}$$_{j}$ + λ $^{i}$λ$^{j}$G$_{ij}$ − (2 λ $^{0}$T$_{2}$ ) $^{2}$GG00 ] . (16) G$_{00}$ − 2 λ $^{j}$G$_{oj}$ + λ $^{i}$λ$^{j}$G$_{ij}$ + (2 λ $^{0}$T$_{2}$ ) $^{2}$det G$_{ij}$ = 0 , (17) G$_{0}$$_{i}$ − λ $^{i}$G$_{ij}$ = 0 . (18) S B = ∫ d $^{3}$σ 4 λ $_{0}$[ G$_{00}$ − (2 λ $^{0}$T$_{2}$ ) $^{2}$det G$_{ij}$ ] . (19) G$_{00}$ + (2 λ $^{0}$T$_{2}$ ) $^{2}$det G$_{ij}$ = 0 , (20) G$_{0}$$_{i}$ = 0 . (21) ds 2 = − dt 2 + 1 64 $^{[}$dψ + 3 sin 2 µ ( d ˜ ψ + cos ˜ θd ˜ φ ) + 2 cos θdφ ] 2 + 1 $_{8}$( dθ 2 + sin 2 θdφ $^{2}$) + 3 4 $^{[}$dµ 2 + 1 4 sin 2 µ $^{(}$d ˜ θ 2 + sin 2 ˜ θd ˜ φ 2 + cos 2 µ ( d ˜ ψ + cos ˜ θd ˜ φ ) 2 )] . (22) ψ = ζτ, ˜ ψ = ν$_{1}$τ, ˜ φ = ν$_{2}$τ, φ = ωτ. (23)