  We describe certain $C^*$-algebras generated by Toeplitz operators with
nilpotent symbols and acting on a poly-Bergman type space of the Siegel domain
$D_{2} \subset \mathbb{C}^{2}$.
  Bounded measurable functions of the form $c(\text{Im}\, \zeta_{1},
\text{Im}\, \zeta_{2} - |\zeta_1|^{2})$ are called nilpotent symbols.
  In this work we consider symbols of the form
  $a(\text{Im}\, \zeta_1) b(\text{Im}\, \zeta_2 -|\zeta_1|^{2})$, where both
limits
  $\lim\limits_{s\rightarrow 0^+} b(s)$ and $\lim\limits_{s\rightarrow +\infty}
b(s)$ exist,
  and $a(s)$ belongs to the
  set of piece-wise continuous functions on
$\overline{\mathbb{R}}=[-\infty,+\infty]$ and
  having one-side limit values at each point of a finite set $D\subset
\mathbb{R}$.
  We prove that the $C^*$-algebra generated by all Toeplitz operators $T_{ab}$
is
  isomorphic to $C(\overline{\Pi})$, where
$\overline{\Pi}=\overline{\mathbb{R}} \times \overline{\mathbb{R}}_+$
  and $\overline{\mathbb{R}}_+=[0,+\infty]$.
