  In the present paper we study the singular pure braid group $SP_{n}$ for
$n=2, 3$. We find generators, defining relations and the algebraical structure
of these groups. In particular, we prove that $SP_{3}$ is a semi-direct product
$SP_{3} = \widetilde{V}_3 \leftthreetimes \mathbb{Z}$, where $\widetilde{V}_3$
is an HNN-extension with base group $\mathbb{Z}^2 * \mathbb{Z}^2$ and cyclic
associated subgroups. We prove that the center $Z(SP_3)$ of $SP_3$ is a direct
factor in $SP_3$.
