  The effect of surface exchange anisotropies is known to play a important role
in magnetic critical and multicritical behavior at surfaces. We give an exact
analysis of this problem in d=2 for the O(n) model by using Coulomb gas,
conformal invariance and integrability techniques. We obtain the full set of
critical exponents at the anisotropic special transition--where the symmetry on
the boundary is broken down to O(n_1)xO(n-n_1)--as a function of n_1. We also
obtain the full phase diagram and crossover exponents. Crucial in this analysis
is a new solution of the boundary Yang-Baxter equations for loop models. The
appearance of the generalization of Schramm-Loewner Evolution SLE_{\kappa,\rho}
is also discussed.
