  We present an N=2-supersymmetric mechanical system whose bosonic sector, with
two degrees of freedom, stems from the reduction of an SU(2) Yang-Mills theory
with the assumption of spatially homogeneous field configurations and a
particular ansatz imposed on the gauge potentials in the dimensional reduction
procedure. The Painleve test is adopted to discuss integrability and we focus
on the role of supersymmetry and parity invariance in two space dimensions for
the attainment of integrable or chaotic models. Our conclusion is that the
relationships among the parameters imposed by supersymmetry seem to drastically
reduce the number of possibilities for integrable interaction potentials of the
mechanical system under consideration.
