  We find $(N+1)/2$ distinct classes (``generations'') of kink solutions in an
$SU(N)\times Z_2$ field theory. The classes are labeled by an integer $q$. The
members of one class of kinks will be globally stable while those of the other
classes may be locally stable or unstable. The kink solutions in the $q^{th}$
class have a continuous degeneracy given by the manifold $\Sigma_q=H/K_q$,
where $H$ is the unbroken symmetry group and $K_q$ is the group under which the
kink solution remains invariant. The space $\Sigma_q$ is found to contain
incontractable two spheres for some values of $q$, indicating the possible
existence of certain incontractable spherical structures in three dimensions.
We explicitly construct the three classes of kinks in an SU(5) model with
quartic potential and discuss the extension of these ideas to magnetic monopole
solutions in the model.
