  Let G be a simple algebraic group defined over an algebraically closed field
of characteristic 0 or a good prime for G. Let U be a maximal unipotent
subgroup of G and \u its Lie algebra. We prove the separability of orbit maps
and the connectedness of centralizers for the coadjoint action of U on (certain
quotients of) the dual \u* of \u. This leads to a method to give a
parametrization of the coadjoint orbits in terms of so-called minimal
representatives which form a disjoint union of quasi-affine varieties.
Moreover, we obtain an algorithm to explicitly calculate this parametrization
which has been used for G of rank at most 8, except E8.
  When G is defined and split over the field of q elements, for q the power of
a good prime for G, this algorithmic parametrization is used to calculate the
number k(U(q), \u*(q)) of coadjoint orbits of U(q) on \u*(q). Since k(U(q),
\u*(q)) coincides with the number k(U(q)) of conjugacy classes in U(q), these
calculations can be viewed as an extension of the results obtained in our
earlier paper. In each case considered here there is a polynomial h(t) with
integer coefficients such that for every such q we have k(U(q)) = h(q).
