$P_{\mu}$ and $Q_{\mu}$ is defined as: $P \cdot Q = \eta_{\mu\nu}P^{\mu}Q^{\nu}=P_{0}Q_{0}-P_{i}Q_{i}$ where the Greek indices $\mu,\nu,\lambda.....=0,1$ correspond to the 2$D$ spacetime directions and the Latin indices $i,j,k....=1$ stand for the space direction [*only*]{}. Our choice of the Levi-Civita tensor $\varepsilon_{\mu\nu}$ is such that $\varepsilon_{01}=+1=\varepsilon^{10}$ and $\varepsilon_{\mu\nu}\varepsilon^{\mu\nu}\,=\,+\,2\,!$, $\varepsilon_{\mu\nu}\varepsilon^{\nu\lambda}\,=\,\delta_{\mu}^{\lambda}$, $\varepsilon_{\mu\nu}\varepsilon^{\mu\lambda}\,=\,-\,\delta_{\nu}^{\lambda}$, etc. The notations for the scalar and superscalar fields have been chosen to be $\phi(x)$ and $\Phi(x,\theta,\bar{\theta})$ on the 2$D$ Minkowskian spacetime manifold and (2, 2)-dimensional supermanifold, respectively, for all the Abelian models under consideration (wherever these fields are required for discussions).

Preliminaries: (Anti-)dual-BRST Symmetries
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To begin with, we discuss here the nilpotent $(s_{(a)d}^2 = 0)$ and absolutely anticommuting $(s_{d}s_{ad}+s_{ad}s_{d} = 0)$ (anti-)dual BRST symmetries $s_{(a)d}$ in the Lagrangian formulation for the 1$D$ rigid rotor which is described by the following first-order Lagrangian (see, e.g.\[16\]) $$\begin{aligned}
L_B = \dot r \, p_r + \dot\vartheta \, p_{\vartheta} - \frac{p_{\vartheta}^2}
{2 r^2} - \lambda \,(r - a) + B\,(\dot \lambda - p_r)
+ \frac{B^2}{2} - i\, \dot{\bar C}\,\dot C + i\, \bar C\, C,\end{aligned}$$ where the pair $(\dot r, \dot{\vartheta})$ is the generalized velocities corresponding to the generalized polar coordinates $(r,\vartheta)$ of the rigid rotor. We have taken the unit mass $(m = 