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The complex Lie superalgebras $\mathfrak{g}$ of type $D(2,1;a)$ - alsodenoted by $\mathfrak{osp}(4,2;a) $ - are usually considered for "non-singular"values of the parameter $a$, for which they are simple. In this paper weintroduce five suitable integral forms of $\mathfrak{g}$, that are well-definedat singular values too, giving rise to "singular specializations" that are nolonger simple: this extends the family of simple objects of type $D(2,1;a)$ infive different ways. The resulting five families coincide for general values of$a$, but are different at "singular" ones: here they provide non-simple Liesuperalgebras, whose structure we describe explicitly. We also perform theparallel construction for complex Lie supergroups and describe their singularspecializations (or "degenerations") at singular values of $a$. Although onemay work with a single complex parameter $a$, in order to stress the overall$\mathfrak{S}_3$-symmetry of the whole situation, we shall work (followingKaplansky) with a two-dimensional parameter $\boldsymbol{\sigma} =(\sigma_1,\sigma_2,\sigma_3)$ ranging in the complex affine plane $\sigma_1 +\sigma_2 + \sigma_3 = 0$.

Singular Degenerations of Lie Supergroups of Type $D(2,1;a)$