The analysis of the linearized fluctuations is in solving the equations one by one in the right order. Recall that one expects to find unfolded equations for two fields: one-form ω ω = ω ω μ μ ( Y | x ) d x μ μ and zero-form C = C ( Y | x ) . From the fourth equation it follows that b does not depend on the auxiliary Z-direction. Therefore, one can identify b = C ( Y | x ) . The second equation then immediately leads to