Using the Atiyah-Ward construction, we examine the solutions of the self-dual Yang-Mills equations for an SU (2) gauge theory, dimensionally reduced from | R(^4) to R(^2). There are two main reasons for doing this: (i) To provide a large class of relatively simple examples which elucidate how non-singularity and physical field configurations are related to the parameterization of the Atiyah-Ward construction. (ii) To construct analogues, for pure non-abelian gauge theories, of the superconducting vortex solutions of the abelian Higgs model, in the hope that these will provide the dominant field configurations describing the QCD vacuum. First, Băcklund transformations are used to construct axially symmetric solutions, and the analogues of the ’t Hooft instantons. These results are then generalised, within the twister theoretic framework of the Atiyah-Ward construction, to produce an infinite dimensional parameter space of complex non-singular solutions in each of the Atiyah-Ward ansătze. The field configurations are expressible as unitary group integrals occurring in lattice gauge theories - this leads to a simple proof of non-singularity, and a convenient means of calculating properties of the field configurations using strong and weak coupling expansions. The structure of the field configurations is further elucidated using symmetry arguments and numerical computations. Finally, suggestions are made as to how these solutions may play a role in the QCD confinement mechanism.