  The SU(2) gauge invariant Dirac-Yang-Mills mechanics of 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 adapted basis the pure
gauge degrees of freedom automatically drop out from the consideration after
projection of the model onto 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.
