remains invariant because the above Lagrangian density transforms to the total spacetime derivatives (see, e.g. \[19\] for details).

Finally, we concentrate on a theoretically interesting system of the Abelian 1-form model of the 2$D$ self-dual bosonic field theory which is described by the following (anti-)BRST invariant Lagrangian density (see, e.g. \[20\] for details) $$\begin{aligned}
\mathcal {L}_{B}^{(s)}\,&=&\,\frac{1}{2}\,\dot{\phi}^{2}\,-\,\frac{1}{2}\,
\dot{v}^{2}\,+\,\dot{v}\,(v'\,-\,\phi')\,+\,\lambda\,[\dot{\phi}\,-\,\dot{v}\,
+\, v'\,-\,\phi']\nonumber\\
& - &\,\frac{1}{2}\,(\phi'\,-\,v')^{2} - \, \frac{1}{2}\,(\dot{\lambda}\,-\,v\,-\,\phi)^2\,
-\,i\,\dot{\bar{C}}\,\dot{C}\,+\,2\,i\,\bar{C}\,C,\end{aligned}$$ where an overdot on fields (e.g. $\dot v\,=\,\partial v/\partial t,\,\dot\phi\,=\,\partial\phi/\partial t$) corresponds to the expression for the “generalized" velocities (where a derivative with respect to the evolution parameter $t$ is taken into account) and the prime on the fields ($\phi'\,=\,\partial\phi/\partial x,\,v'\,=\,\partial v/\partial x$) is the space derivative with respect to the space coordinate $x$. Here $v(x)$ field is the Wess-Zumino (WZ) field and $\phi(x)$ field is the 2$D$ self-dual bosonic field (of our present 2$D$ self-dual field theory). Rest of the symbols have already been explained earlier. The above Lagrangian density (8) is endowed with the following (anti-)dual-BRST symmetry transformations ($s_{(a)d}$) $$\begin{aligned}
&&s_{ad}\,\lambda\,=\,C,\qquad s_{ad}\,\phi\,=\,\frac{\dot C}{2},\qquad s_{ad}\,v\,=\,\frac{\dot C}{2},
\qquad \quad s_{ad}\,C\,=\,0,\\\nonumber
&&s_{ad}\,\bar C\,=\,\frac{i}{2}\,(\dot \phi\,-\,\dot v\,+\,v'\,-\,\phi'),\quad
s_{ad}\,(\dot\phi\,-\,\dot v\,+\,v'\,-\,\phi')\,=\,0, \\ \nonumber
&&s_{d}\,\lambda\,=\,\bar C, \qquad \quad s_{d}\,\phi\,=\,\frac{\dot{\bar C}} {2}, \qquad \quad
s_{d}\,v\,=\,\frac{\dot {\bar C}}{2}, \quad \qquad s_{d}\,\bar C\,=\,0,\\ 