  We study subgroups of ${\rm PU}(2,1)$ generated by two non-commuting
unipotent maps $A$ and $B$ whose product $AB$ is also unipotent. We call
$\mathcal{U}$ the set of conjugacy classes of such groups. We provide a set of
coordinates on $\mathcal{U}$ that make it homeomorphic to $\mathbb{R}^2$ . By
considering the action on complex hyperbolic space $\mathbf{H}^2_{\mathbb{C}}$
of groups in $\mathcal{U}$, we describe a two dimensional disc ${\mathcal Z}$
in $\mathcal{U}$ that parametrises a family of discrete groups. As a corollary,
we give a proof of a conjecture of Schwartz for $(3,3,\infty)$-triangle groups.
We also consider a particular group on the boundary of the disc ${\mathcal Z}$
where the commutator $[A,B]$ is also unipotent. We show that the boundary of
the quotient orbifold associated to the latter group gives a spherical CR
uniformisation of the Whitehead link complement.
