Renormalized critical behavior or first-order phase transitions?

The exactly solvable Baker-Essam model for a compressible Ising lattice is re-examined. We find that, depending on the type of constraint imposed upon the system, the second-order phase transition either gets renormalized or is changed into a first-order transition. We resolve the apparent disagreement of this result with Fisher's renormalization theory by showing that, in fact, our result is typical of the correct state of affairs to be expected on the basis of the general theory of critical points in systems with hidden variables.