On the hamiltonicity problem of bicirculants: a reduction to cyclic Haar graphs

A bicirculant is a regular graph that admits an automorphism having two vertex-orbits of the same size. A bicirculant can be described as follows. Given an integer $m \ge 1$ and sets $R, S, T \subseteq \mathbb Z_m$ such that $R=-R$, $T=-T$, $0 \not\in R \cup T$ and $0 \in S$, the graph $B(m;R,S,T)$ has vertex set $V=\{u_0,\dots,u_{m-1},v_0,\dots,v_m-1\}$ and edge set $E=\{u_iu_{i+j}| \ i \in\mathbb Z_m, j \in R\} \cup \{v_iv_{i+j}| \ i \in\mathbb Z_m, j \in T\} \cup\{u_iv_{i+j}| \ i \in\mathbb Z