Topological and non-topological kink families in non-linear $({\mathbb{S}}^1\times {\mathbb{S}}^1)$-Sigma models

In this paper we construct a family of Hamilton-Jacobi separable non-linear ${\mathbb{S}}^1\times {\mathbb{S}}^1$ Sigma models for which the kink variety can be analytically identified and for which the linear stability of the emerging kinks is ensured. Furthermore, a model with only one vacuum point is found, where all kinks are forced to be non-topological. The non-simply connectedness of the torus guarantees the global stability of all the non-topological kinks in these models.