  We present a self-dual non-Abelian N=1 supersymmetric tensor multiplet in
D=2+2 space-time dimensions. Our system has three on-shell multiplets: (i) The
usual non-Abelian Yang-Mills multiplet (A_\mu{}^I, \lambda{}^I) (ii) A
non-Abelian tensor multiplet (B_{\mu\nu}{}^I, \chi^I, \varphi^I), and (iii) An
extra compensator vector multiplet (C_\mu{}^I, \rho^I). Here the index I is for
the adjoint representation of a non-Abelian gauge group. The duality symmetry
relations are G_{\mu\nu\rho}{}^I = - \epsilon_{\mu\nu\rho}{}^\sigma
\nabla_\sigma \varphi^I, F_{\mu\nu}{}^I = + (1/2)
\epsilon_{\mu\nu}{}^{\rho\sigma} F_{\rho\sigma}{}^I, and H_{\mu\nu}{}^I =
+(1/2) \epsilon_{\mu\nu}{\rho\sigma} H_{\rho\sigma}{}^I, where G and H are
respectively the field strengths of B and C. The usual problem with the
coupling of the non-Abelian tensor is avoided by non-trivial Chern-Simons terms
in the field strengths G_{\mu\nu\rho}{}^I and H_{\mu\nu}{}^I. For an
independent confirmation, we re-formulate the component results in superspace.
As applications of embedding integrable systems, we show how the {\cal N} = 2,
r = 3 and {\cal N} = 3, r = 4 flows of generalized Korteweg-de Vries equations
are embedded into our system.
