1)$ while defining the pair $(p_{r}, p_{\vartheta})$ as the conjugate momenta corresponding to the coordinates $(r, \vartheta)$. Here $\lambda(t)$ is the “gauge" variable of the theory (which is a 1-form ${\lambda}^{(1)} = dt\;\lambda (t)$ on a 1$D$ manifold) and $B(t)$ is the Nakanishi-Lautrup type auxiliary variable. The anticommuting $(C(t)\,\bar{C}(t)+\bar{C}(t)\,C(t) = 0)$ fermionic $(C^2 = \bar{C}^2 = 0)$ (anti-)ghost variables $(\bar{C})C$ are required to maintain the unitarity in the theory. All the variables are the function of an evolution parameter $t$ and an overdot ($\dot r, \dot{\vartheta}, \dot {\lambda}, \dot C, \dot{\bar{C}},$ etc.) corresponds to a single derivative (i.e.$\dot r = dr/dt,\, \dot v = dv/dt,$etc.) with respect to $t$. It can be readily checked \[16\] that, under the following (anti-)dual BRST symmetry transformations $(s_{(a)d})$ $$\begin{aligned}
&&s_{ad}\,\lambda = C, \quad s_{ad}\,p_{r}\,=\,\dot C, \quad s_{ad}\,\bar C = -\;i\;(r-a),\quad
s_{ad}\,[r, \vartheta, B, p_{\vartheta}, C]\,=\,0, \\ \nonumber
&&s_{d}\,\lambda = \bar C, \quad s_{d}\,p_{r}\,=\,\dot{\bar C}, \quad\quad s_{d}\,C = i\;(r-a),\quad \quad
s_{d}\,[r, \vartheta, B, p_{\vartheta}, \bar C]\,=\,0, \end{aligned}$$ the Lagrangian (1) and gauge-fixing term remain invariant $(s_{(a)d} L_B = 0, s_{(a)d} (\dot\lambda - p_r) = 0)$.

We now focus our attention on the (anti-)dual BRST symmetry transformations for the modified version of 2$D$ Proca theory (with 