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