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In this paper the whole kink varieties arising in several massive non-linearSigma models whose target space is the torus ${\mathbb S}^1\times{\mathbb S}^1$are analytically calculated. This possibility underlies the construction offirst-order differential equations by adapting the Bogomolny procedure tonon-Euclidean spaces. Among the families of solutions non-topological kinksconnecting the same vacuum are found. This class of solutions are usuallyconsidered to be not globally stable. However, in this context the topologicalconstraints obtained by the non-simply connectedness of the target space turnthese non-topological kinks into globally stable solutions. The analyticalresolution of the equations allows the complete study of the linear stabilityfor some basic kinks.

Kinks in massive non-linear ${\mathbb S}^1\times{\mathbb S}^1$-Sigma models