  Let $SB_n$ be the singular braid group generated by braid generators
$\sigma_i$ and singular braid generators $\tau_i$, $1 \leq i \leq n-1$. Let
$ST_n$ denote the group that is the kernel of the homomorphism that maps, for
each $i$, $\sigma_i$ to the cyclic permutation $(i, i+1)$ and $\tau_i$ to $1$.
In this paper we investigate the group $ST_3$. We obtain a presentation for
$ST_3$. We prove that $ST_3$ is isomorphic to the singular pure braid group
$SP_3$ on $3$ strands. We also prove that the group $ST_3$ is semi-direct
product of a subgroup $H$ and an infinite cyclic group, where the subgroup $H$
is an HNN-extension of ${\mathbb Z}^2 \ast {\mathbb Z}^2$.
