  Let $(X,\sigma,J)$ be a compact K\"{a}hler Calabi-Yau manifold equipped with
a symplectic circle action. By Frankel's theorem \cite{F}, the action on $X$ is
non-Hamiltonian and $X$ does not have any fixed point. In this paper, we will
show that a symplectic circle action on a compact non-K\"{a}hler symplectic
Calabi-Yau manifold may have a fixed point. More precisely, we will show that
the symplectic $S^1$-manifold constructed by D. McDuff \cite{McD} has the
vanishing first Chern class. This manifold has the Betti numbers $b_1 = 3$,
$b_2 = 8$, and $b_3 = 12$. In particular, it does not admit any K\"{a}hler
structure.
