  We establish the equivalence of a recently introduced discrete model of the
hydrophobic interaction, as well as its extension to continuous state
variables, with the Ising model in a magnetic field with temperature-dependent
strength. In order to capture the effect of symmetries of the solvent particles
we introduce a generalized multi-state model. We solve this model - which is
not of the Ising type - exactly in one dimension. Our findings suggest that a
small increase in symmetry decreases the amplitude of the solvent-mediated part
of the potential of mean force between solute particles and enhances the
solubility in a very simple fashion. High symmetry decreases also the range of
the attractive potential. This weakening of the hydrophobic effect observed in
the model is in agreement with the notion that the effect is entropic in
origin.
