  We analyze the interrelation between the coexistence curve of the
Lennard-Jones fluid and the Ising model in two and three dimensions within the
global isomorphism approach proposed earlier [V. L. Kulinskii, J. Phys. Chem. B
\textbf{114} 2852 (2010)]. In case of two dimensions we use the exact Onsager
result to construct the binodal of the corresponding Lennard-Jones fluid and
compare it with the results of the simulations. In the three dimensional case
we use available numerical results for the Ising model for the corresponding
mapping. The possibility to observe the singularity of the binodal diameter is
discussed.
