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We show that scale invariant scattering theory allows to exactly determinethe critical points of two-dimensional systems with coupled $O(N)$ and Isingorder pameters. The results are obtained for $N$ continuous and includecriticality of loop gas type. In particular, for $N=1$ we exhibit threecritical lines intersecting at the Berezinskii-Kosterlitz-Thouless transitionpoint of the Gaussian model and related to the $Z_4$ symmetry of the isotropicAshkin-Teller model. For $N=2$ we classify the critical points that can arisein the XY-Ising model and provide exact answers about the critical exponents ofthe fully frustrated XY model.

Critical points of coupled vector-Ising systems. Exact results