In this 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 = V ̃ 3 ⋋ ℤ , where V ̃ 3 is an HNN-extension with base group ℤ 2 ∗ ℤ 2 and cyclic associated subgroups. We prove that the center Z ( SP 3) of SP 3 is a direct factor in SP 3 .