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.