4 Gauge invariance and dual equivalence of Abelian and non-Abelian actions via dual embedding formalism is dedicated to the calculation of G ( a$_{i}$, p$_{i}$, η ). In order to begin with the first step, we impose that this new symplectic tensor ( ˜ f $^{(0)}$) has a zero-mode ˜ ν , consequently, we obtain the following condition with where the arbitrary function G ( a$_{i}$, p$_{i}$, η ) is expressed as an expansion in terms of the WZ field, given by At this point, f becomes degenerated and in consequence, we introduce an obstruction to solve the Hamiltonian equation of motion given by equation (2.8). Assuming that the zero-mode ˜ ν (0) ˜ θ is and satisfies the following boundary condition where The symplectic variables were extended to also encom- pass the WZ variable ˜ ξ (0) ˜ θ = ( ξ (0) $^{θ}$, η ) (with ˜ θ = 1 , 2 , . . . , 2 N + 1) and the first-iterative symplectic poten- tial becomes and using the relation given in (2.17) together with (2.16), we have a system of equations, where In this context, the new canonical momenta are The matrix elements µ θ are chosen in order to disclose the desired gauge symmetry. Note that in this formal- ism the zero-mode ˜ ν (0) ˜ θ is the gauge symmetry genera- tor. It is worth to mention that this feature is impor- tant because it opens up the possibility to disclose the desired hidden gauge symmetry from the noninvariant model. From relation (2.17) some differential equations involving Ψ( a$_{i}$, p$_{i}$ ) are obtained, i. e., the equation (2.19), and after a straightforward computation, Ψ( a$_{i}$, p$_{i}$ ) can be determined. and the new symplectic tensor is given by that is In order to compute G ( a$_{i}$, p$_{i}$, η ) following the second step of the method, it is mandatory that no constraints arise from the contraction of the zero-mode (˜ ν (0) ˜ $^{θ}$) with the gradient of the potential ˜ V $^{(0)}$( a$_{i}$, p$_{i}$, η ). This condi- tion generates a general differential equation, which reads as To sum up, basically, we have two steps: the first one is addressed to compute Ψ( a$_{i}$, p$_{i}$ ) while the second one that allows us to compute all the correction terms G ( n $^{)}$( a$_{i}$, p$_{i}$, η ) as functions of η . Notice that this poly- nomial expansion in terms of η is equal to zero. Conse- quently, all the coefficients for each order in η must be identically null. Given this, each correction term as func- tion of η is determined. For a linear correction term, we ˜ ν (0) ˜ θ ∂ ˜ V $^{(0)}$( a$_{i}$, p$_{i}$, η ) ∂ ˜ ξ (0) ˜ θ = 0 , (2.21) µ θ ∂V $^{(0)}$( a$_{i}$, p$_{i}$ ) ∂ξ (0) θ + µ θ ∂ G $^{(1)}$( a$_{i}$, p$_{i}$, η ) ∂ξ (0) θ + µ θ ∂ G $^{(2)}$( a$_{i}$, p$_{i}$, η ) ∂ξ (0) θ + . . . + ∂ G $^{(1)}$( a$_{i}$, p$_{i}$, η ) ∂η + ∂ G $^{(2)}$( a$_{i}$, p$_{i}$, η ) ∂η + · · · = 0 , (2.22) ˜ f (0) θ ˜ β = ( f (0) θβ f (0) θη f (0) ηβ 0 ) . (2.16) ˜ f (0) θ ˜ β = ∂ ˜ A (0) ˜ β ∂ ˜ ξ (0) ˜ θ − ∂ ˜ A (0) ˜ θ ∂ ˜ ξ (0) ˜ β , (2.15) ˜ A (0) ˜ θ = { A (0) θ , with ˜ θ =1,2,. . . ,2N Ψ , with ˜ θ = 2N + 1 (2.14) ˜ V $^{(0)}$( a$_{i}$, p$_{i}$, η ) = V $^{(0)}$( a$_{i}$, p$_{i}$ ) + ∞ ∑ n =1 G ( n $^{)}$( a$_{i}$, p$_{i}$, η ) . (2.13) f (0) ηβ = ∂A (0) β ∂η − ∂ Ψ ∂ξ (0) β . (2.20) µ $^{θ}$f (0) θβ + f (0) ηβ = 0 , (2.19) G ( a$_{i}$, p$_{i}$, η = 0) = 0 . (2.12) G ( n $^{)}$( a$_{i}$, p$_{i}$, η ) ∼ η n , G ( a$_{i}$, p$_{i}$, η ) = ∞ ∑ n =1 G ( n $^{)}$( a$_{i}$, p$_{i}$, η ) , , (2.11) ˜ ν (0) ˜ θ ˜ f (0) θ ˜ β = 0 . (2.17) ˜ V (0) = V (0) + G ( a$_{i}$, p$_{i}$, η ) , (2.10)