in a definite and precise manner. It is worthwhile to mention that the condition $B_{1}\,+\,{\bar B}_{2}\,=\,0$ is the [*trivial*]{} CF-type condition. This restriction is a [*physical*]{} condition in our theory because it is an (anti-)co-BRST invariant quantity.

We resort to the additional restrictions on the supervariables that are motivated by the basic requirements of AVSF which state that the (anti-)co-BRST invariant quantities should be independent of the “soul" coordinates. In this connection, we observe the following $$\begin{aligned}
s_{(a)d}\,\bigl[ \dot r\,p_r - i\, \dot{\bar C}\, \dot C\bigr] = 0, \qquad\qquad s_{(a)d}\,\bigl[ \lambda\,(r - a)  
- i\, \bar{C}\, C\bigr] = 0,\end{aligned}$$ which, ultimately, imply the following equalities due to DGIRs, namely; $$\begin{aligned}
&&\dot{R} (t, \theta, \bar\theta) \, P^{(R)}_{r} (t, \theta, \bar\theta)
-i\,\dot{\bar{F}}^{(R)} (t, \theta, \bar\theta)\, 
\dot{F}^{(R)} (t, \theta, \bar\theta) = \dot{r}\,p_{r} - i\,\dot{\bar{C}}\,\dot{C},\\ \nonumber
&&\Lambda (t, \theta, \bar\theta) \,[R (t, \theta, \bar\theta) - a] 
- i\,\bar{F}^{(R)} (t, \theta, \bar\theta) \,F^{(R)} (t, \theta, \bar\theta)  = \,\lambda\,(r - a) - i\, \bar{C}\,C,\end{aligned}$$ where the [*new*]{} notations (with $R (t, \theta, \bar\theta) = r( t)$) are explicitly written as $$\begin{aligned}
&& F^{(R)}(t, \theta, \bar{\theta}) = C(t) + i\,\bar\theta\, {\cal B},\qquad
{\bar {F}}^{(R)}(t, \theta, \bar{\theta}) = \bar C(t) - i\,\theta\, {\cal B},\nonumber\\
&& P^{(R)} (t, \theta, 