\,+\,\theta\bar\theta\,[-\,i \,\varepsilon_{\mu\nu}\,\partial^{\nu}{\cal B}(x)]\nonumber\\
&\equiv& \,A_{\mu}(x)\,+\,\theta\,
\bigl(s_{ad}\,A_{\mu}(x)\bigr)\,+\,\bar \theta \,
\bigl(s_{d}\,A_{\mu}(x)\bigr) 
\,+\,\theta\bar\theta\,\bigl(s_{d}\,s_{ad}\, A_{\mu}(x)\bigr) .\end{aligned}$$ It is very clear that we have derived the following (anti-)co-BRST symmetry transformations for the fields ($A_\mu (x), C (x), \bar C (x)$) due to superfield formalism: $$\begin{aligned}
&&s_{d}\,A_{\mu}\,=\,-\,\varepsilon_{\mu\nu}\,\partial^{\nu}\bar C,\quad 
s_{ad}\,A_{\mu}\,=\,-\,\varepsilon_{\mu\nu}\,\partial^{\nu} C,
\quad s_{d}s_{ad}\,A_{\mu}\,=\,-\,i\,\varepsilon_{\mu\nu}\,\partial^{\nu}\cal B, \nonumber \\
&&s_{d}\,C\,=\,-\,i\,{\cal B}, \qquad s_{ad}\,C\,=\,0, \qquad s_{d}s_{ad}\,C\,=\,0, 
\qquad \\ \nonumber 
&& s_{d}\,\bar C = 0, 
\qquad s_{ad}\,\bar C = \,i\,{\cal B}, \qquad s_{d}s_{ad}\,\bar C = 0.\end{aligned}$$ The nilpotency and absolute anticommutativity properties of the (anti-)co-BRST symmetry transformations imply that we have $s_{(a)d}\,{\cal B} = 0$. Up to this point, our results are [*same*]{} for the superfield description of the modified versions of 2D [*Proca*]{} and [*anomalous gauge theories*]{} because the above transformations are common for both the theories.

To determine the (anti-)dual-BRST symmetry transformations for the $\tilde \phi(x)$ field of the modified version of the 2$D$ Proca theory (cf. Eq. (3)), we observe that $s_{(a)d}\,\bigl[A_{\mu} - \frac {1}{m}\,\varepsilon_{\mu\nu}\,\partial^{\nu}{\tilde \phi}\bigr] = 0$. Thus, according to the basic requirements of AVSF, we demand that this quantity should remain independent of the “soul" coordinates, namely; $$\begin{aligned}
{\cal B}_{\mu}^{(d)}(x, \theta, \bar\theta) - \frac {1}{m}\,\varepsilon_{\mu\nu}\,\partial^{\nu}{\tilde \Phi}
(x, \theta, \bar\theta) = 
A_{\mu} (x) - \frac {1}{m}\,\varepsilon_{\mu\nu}\,\partial^{\nu}{\tilde \phi}(x).\end{aligned}$$ Now 