$\iota$ be a involution on $S^1=\mathbb{R}/2\pi\mathbb{Z}$ defined by $\iota(\theta)=\theta+\pi$. If $(\L,\L_\bullet)$ is a Stokes structure, $\iota^{-1}\L$ has a natural Stokes filtration by $$\begin{aligned}
\label{involution}
           (\iota^{-1}\L)_{\leqslant c,\theta}:=\L_{\leqslant -c,\theta+\pi} \ (c\in {\mathbb{C}},\ \theta \in S^1).
           \end{aligned}$$ The graded quotient ${\mathrm{gr}}_c(\iota^{-1}\L)$ is naturally identified with $\iota^{-1}{\mathrm{gr}}_{-c}\L$ ([@herexa §3.a]). The dual local system $\L^\vee={\mathscr{H}om}(\L,{\bm k}_{S^1})$ also has a natural filtration defined by $$\begin{aligned}
\label{vee}
            (\L^\vee)_{\leqslant c}:=(\L_{<-c})^\perp
           \end{aligned}$$ where $(\L_{<c})^\perp$ consists of local morphisms $\L\to {\bm k}_{S^1}$ which send $\L_{<c}$ to $0$. The graded quotient ${\mathrm{gr}}_c(\L^\vee)$ is naturally identified with $({\mathrm{gr}}_{-c}\L)^\vee$. Hence ${\mathbb{D}}\L=\iota^{-1}\L^\vee$ has natural Stokes filtration by $$\begin{aligned}
\label{duality for Stokes filtration}
            ({\mathbb{D}}\L)_{\leqslant c}=(\L_{<c,\theta+\pi})^\perp\ (c\in{\mathbb{C}}, \theta\in S^1),
           \end{aligned}$$ whose graded quotient ${\mathrm{gr}}_c({\mathbb{D}}\L)$ is naturally identified with ${\mathbb{D}}({\mathrm{gr}}_c\L)$.

The category of Stokes filtered local systems on $S^1$ defined above is denoted by ${\sf St}_{\bm{k}}(S^1)$. In this paper we only consider Stokes filtered local systems on $S^1$. We abbreviate $S^1$ and simply denote it by ${\sf St}_{\bm{k}}$. For each $(\L,\L_\bullet)\in {{\sf St}}_{\bm{k}}$, the set $\{c\in {\mathbb{C}}\mid  {\mathrm{gr}}_c \L\neq  0\}$ is called exponents of $(\L,\L_\bullet)$. For a finite subset $C$ in ${\mathbb{C}}$, ${\sf St}_{\bm{k}}^C$ denote the full subcategory of Stokes filtered local systems whose exponents are contained in $C$. We remark that ${\mathbb{D}}$ defines a contravariant functor 