  Associated to a convex integral polygon $N$ in the plane are two integrable
systems: the cluster integrable system of Goncharov and Kenyon constructed from
the planar dimer model, and the Beauville integrable system, associated with
the toric surface of $N$. There is a birational map, called the spectral
transform, between the phase spaces of the two integrable systems. When $N$ is
the triangle $\text{Conv}\{(0,0),(d,0),(0,d)\}$, we show that the spectral
transform is a birational isomorphism of integrable systems.
