  We give exact solutions for a recently developed ~$N=1$~ locally
supersymmetric self-dual gauge theories in $~(2+2)\-$dimensions. We give the
exact solutions for an $~SL(2)$~ self-dual Yang-Mills multiplet and what we
call ``self-dual tensor multiplet'' on the gravitational instanton background
by Eguchi-Hanson. We use a general method to get an $~SL(2)$~ self-dual
Yang-Mills solution from any known self-dual gravity solution. Our result is
the first example of exact solutions for the coupled system of these $~N=1$~
locally supersymmetric self-dual multiplets in ~$(2+2)\-$dimensions, which is
supposed to have strong significance for integrable models in lower-dimensions
upon appropriate dimensional reduction or truncation. We also inspect the
consistency of our exact solutions as a background for $~N=2$~ superstring
coupled to the Wess-Zumino-Witten term in $~\s\-$model formulation.
