generators provide the geometrical meaning to the nilpotency ($s_{(a)d}^2 = 0$) and absolute anticommutativity ($s_d\;s_{ad} + s_{ad}\;s_d = 0$) of the (anti-)co-BRST symmetries.

We now focus on the derivation of the (anti-)co-BRST symmetry transformations ($s_{(a)d}$) in the context of the modified versions of the 2$D$ Proca and anomalous Abelian 1-form gauge theories within the framework of AVSF. In this connection, first of all, we observe that the gauge-fixing term ($\partial \cdot A\,\pm \,m\phi$) remains invariant \[i.e. $s_{(a)d}\,(\partial \cdot A \pm m \phi)\,=\,0$\] under $s_{(a)d}$ (because, separately and independently, we have: $s_{(a)d}(\partial \cdot A)\,=\,0, \, s_{(a)d}\phi\,=\,0$). We note that $(\partial \cdot A)$ has its origin in the co-exterior derivative ($\delta$) because $\delta A^{(1)} = -\,\ast d \ast (dx^{\mu}\,A_{\mu})\,=\,(\partial \cdot A)$. Thus, we have to generalize this relationship onto the (2, 2)-dimensional supermanifold parametrized by the superspace co-ordinates ($x^{\mu}, \theta, \bar\theta$). Thus, according to the basic tenets of AVSF, we have the following equality (see, e.g. \[18\] for details) $$\begin{aligned}
\star \,{\tilde d} \star   \tilde A^{(1)}=\,\ast  \,d \ast  A^{(1)},
\qquad \quad\phi(x) \to \Phi(x, \theta, \bar\theta)\,=\,\phi(x),\end{aligned}$$ where $\star$ is the Hodge duality operation on the (2, 2)-dimensional supermanifold and other relevant symbols have already been explained earlier. In our earlier works \[18\], the 