We propose a variant scheme of the Gauge Unfixing formalism which modifies directly the original phase space variables of a constrained system. These new variables are gauge invariant quantities. We apply our procedure in a mixed constrained system that is the Abelian Pure Chern Simons Theory where several gains are obtained. In particular, from the gauge invariant Hamiltonian and using the inverse Legendre transformation, we obtain the same initial Abelian Pure Chern Simons Lagrangian as the

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... uge invariant Lagrangian. This result shows that the gauge symmetry of the action is certainly preserved.