  In this paper, we propose a generalization of an improved gauge unfixing
formalism in order to generate gauge symmetries in the non-Abelian valued
systems. This generalization displays a proper and formal reformulation of
second-class systems within the phase space itself. We then present our
formalism in a manifestly gauge invariant resolution of the $SU(N)$ massive
Yang-Mills and $SU(2)$ Skyrme models where gauge invariant variables are
derived allowing then the achievement of Dirac brackets, gauge invariant
Hamiltonians and first-class Lagrangians.
