  In this paper the whole kink varieties arising in several massive non-linear
Sigma models whose target space is the torus ${\mathbb S}^1\times{\mathbb S}^1$
are analytically calculated. This possibility underlies the construction of
first-order differential equations by adapting the Bogomolny procedure to
non-Euclidean spaces. Among the families of solutions non-topological kinks
connecting the same vacuum are found. This class of solutions are usually
considered to be not globally stable. However, in this context the topological
constraints obtained by the non-simply connectedness of the target space turn
these non-topological kinks into globally stable solutions. The analytical
resolution of the equations allows the complete study of the linear stability
for some basic kinks.
