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transformations. Due to this, all correction terms G ( n ) with n ≥ 3 are zero. The infinitesimal gauge transforma- tions generated by the zero-mode ( δξ$_{i}$ = εν $^{(0)}$) are 

It is easy to verify that the Hamiltonian, identified as be- ing the new symplectic potential ˜ U $^{(0)}$, is invariant under these infinitesimal gauge transformation above, namely, 

At this point, we are interested in the analysis of this result and we would like also to demonstrate that the anomaly was canceled. It will be carried out from Dirac’s point of view. From the Lagrangian (5.28) the chains of primary constraints are computed as, 

These primary constraints will be introduced into the Hamiltonian through Lagrange multipliers. In this way, the primary Hamiltonian, 

Since the constraint ϕ$_{1}$ has no time evolution, the follow- ing secondary constraint is required 

and no more constraints arise from the temporal stability condition. In this way, the total Hamiltonian is 

The time stability condition for the constraint χ$_{1}$ just allows us to determine the Lagrange multiplier λ$_{3}$ . In this way, the gauge invariant version of the model has three constraints, namely, ϕ$_{1}$ , ϕ$_{2}$ and χ$_{1}$ . The corresponding Dirac matrix, given by, 

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is singular. As the Dirac matrix is singular, the model has both first and second-class constraints. Through a constraint combination, we obtain a set of first-class con- straints such as, 

and a set of second-class constraints, given by 

It is again easy to verify that ˜ χ$_{1}$ is a first-class constraint, identified as the Gauss law, while the others are second- class constraints. Note that the anomaly was removed. Hence, the Gauss law is also recognized as being the gen- erator of the gauge transformation given in Eq. (5.30). 

At this stage, we will compute the degrees of free- dom of the gauge invariant model proposed by us. The model has one first-class and two second-class constraints and the phase space dimensions sum eight dependent fields, i.e., ( φ, π$_{φ}$, A$_{1}$, π$_{1}$, A$_{0}$, π$_{0}$, θ, π$_{θ}$ ). The first-class constraint eliminates two fields, while the second-class constraints eliminate two fields, summing then four fields eliminated. Hence, the model has four independent fields, i.e., there are two independent degrees of freedom. 

In order to obtain the Dirac brackets, the set of second- class constraints, ˜ ϕ$_{1}$ and ˜ ϕ$_{2}$ , will be assumed equal to zero in a strong way. After a straightforward computa- tion, the Dirac brackets among the phase space fields are obtained as 

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