  We present a system of a self-dual Yang-Mills field and a self-dual
vector-spinor field with nilpotent fermionic symmetry (but not supersymmetry)
in 2+2 dimensions, that generates supersymmetric integrable systems in lower
dimensions. Our field content is (A_\mu{}^I, \psi_\mu{}^I, \chi^{I J}), where I
and J are the adjoint indices of arbitrary gauge group. The \chi^{I J} is a
Stueckelberg field for consistency. The system has local nilpotent fermionic
symmetry with the algebra \{N_\alpha{}^I, N_\beta{}^J \} = 0. This system
generates supersymmetric Kadomtsev-Petviashvili equations in D=2+1, and
supersymmetric Korteweg-de Vries equations in D=1+1 after appropriate
dimensional reductions. We also show that a similar self-dual system in seven
dimensions generates self-dual system in four dimensions. Based on our results
we conjecture that lower-dimensional supersymmetric integral models can be
generated by non-supersymmetric self-dual systems in higher dimensions only
with nilpotent fermionic symmetries.
