Structure of $4$-strand singular pure braid group

We construct a finite presentation for the singular pure braid group $SP_4$ on $4$ strands. As consequence it was proved that the center $Z(SP_4)$, which is the infinite cyclic group, is a direct factor in $SP_4$. On the other side, we establish that $Z(SP_4)$ is not a direct factor in the singular braid group $SG_4$.