  We consider small ballistic quantum dots weakly coupled to the leads in the
chaotic regime and look for significant spin-orbit effects. We find that these
effects can become quite prominent in the vicinity of degeneracies of many-body
energies. We illustrate the idea by considering a case where the intrinsic
exchange term -JS^2 brings singlet and triplet many-body states near each
other, while an externally tunable Zeeman term then closes the gap between the
singlet and the one of the triplet states (with spin projection parallel the
external field). Near this degeneracy, the spin-orbit coupling leads to a
striking temperature dependence of the conductance, with observable effects of
order unity at temperatures lower than the strength of the spin-orbit coupling.
Under favorable circumstances, spelled out in the paper, these order unity
effects in the conductance persist to temperatures much higher than the
spin-orbit coupling strength. Our conclusions are unaffected by the presence of
non-universal perturbations. We suggest a class of experiments to explore this
regime.
