Descrição: A novel feature of the ${\mathbb Z}_2\times {\mathbb Z}_2$-graded supersymmetry which finds no counterpart in ordinary supersymmetry is the presence of $11$-graded exotic bosons (implied by the existence of two classes of parafermions). Their interpretation, both physical and mathematical, presents a challenge. The role of the ``exotic bosonic coordinate" was not considered by previous works on the one-dimensional \zzg superspace (which was restricted to produce point-particle models). By treating this coordinate at par with the other graded superspace coordinates new consequences are obtained. The graded superspace calculus of the \zzg worldline super-Poincar\'e algebra induces two-dimensional \zzg relativistic models; they are invariant under a new \zzg $2D$ super-Poincar\'e algebra which differs from the previous two \zzg $2D$ versions of super-Poincar\'e introduced in the literature. In this new superalgebra the second translation generator and the Lorentz boost are $11$-graded. Furthermore, if the exotic coordinate is compactified on a circle ${\bf S}^1$, a \zzg closed string with periodic boundary conditions is derived. \\ The analysis of the irreducibility conditions of the $2D$ supermultiplet implies that a larger $(\beta$-deformed, where $\beta\geq 0$ is a real parameter) class of point-particle models than the ones discussed so far in the literature (recovered at $\beta=0$) is obtained. While the spectrum of the $\beta=0$ point-particle models is degenerate (due to its relation with an ${\cal N}=2$ supersymmetry), this is no longer the case for the $\beta> 0$ models.

Número: CBPF-NF-004/22