[en] The SU(2) gauge invariant Dirac---Yang-Mills mechanics of a spatially homogeneous isospinor and gauge fields is considered in the framework of the generalized Hamiltonian approach. The unconstrained Hamiltonian system equivalent to the model is obtained using the gaugeless method of Hamiltonian reduction. The latter includes the Abelianization of the first class constraints, putting the second class constraints into the canonical form and performing a canonical transformation to a set of adapted coordinates such that a subset of the new canonical pairs coincides with the second class constraints and part of the new momenta is equal to the Abelian constraints. In the adapter basis the pure gauge degrees of freedom automatically drop out from the consideration after projection of the model into the constraint shell. Apart from the elimination of these ignorable degrees of freedom a further Hamiltonian reduction is achieved due to the three-dimensional group of rigid symmetry possessed by the system