Let G be a non-elementary subgroup of SL₂(Z). If m is a probability measure on T² which is G-invariant, then m is a convex combination of the Haar measure and an atomic probability measure supported by rational points. The same conclusion holds under the weaker assumption that m is p-stationary, i.e. m=p*m, where p is a finitely supported probability measure on G whose support generates G. The approach works more generally for G < SL_d(Z).