Restoring the gauge invariance in non-Abelian second-class theories

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.