Competing interactions in the XYZ model

We study the interplay between an $\mathrm{XY}$ anisotropy $\ensuremath{\gamma},$ exchange modulations, and an external magnetic field along the z direction in an $\mathrm{XYZ}$ chain using bosonization and Lanczos diagonalization techniques. We find an Ising critical line in the space of couplings which occur due to competing relevant perturbations that are present. More general situations are also discussed.