Singular degenerations of Lie supergroups of type D(2,1;a)

The complex Lie superalgebras $ \frak{g} $ of type $ D(2,1;a) $ --- also denoted by $ \frak{osp}(4,2;a) $ --- are usually considered for ``non-singular'' values of the parameter $ a \, $, for which they are simple. In this paper we introduce five suitable integral forms of $\frak{g} \, $, that are well-defined at singular values too, giving rise to ``singular specializations'' that are no longer simple: this extends the family of {\it simple} objects of type $ D(2,1;a) $ in five different ways. The resulting five families coincide for general values of $ a \, $, but are different at ``singular'' ones: here they provide non-simple Lie superalgebras, whose structure we describe explicitly. We also perform the parallel construction for complex Lie supergroups and describe their singular specializations (or ``degenerations'') at singular values of $ a \, $. Although one may work with a single complex parameter $ a \, $, in order to stress the overall $ \frak{S}_3 $--symmetry of the whole situation, we shall work (following Kaplansky) 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 \, $.