The Clifford index of an algebraic curve behaves quite interestingly when the curve lies on a K3 surface. For example, it does not change if we let the curve move in its linear system. Moreover, if the curve is non-general in moduli its Clifford index is always cut out by a linear system on the surface. We will discuss a possible classification of these linear systems when the ambient K3 surface carries non-symplectic automorphisms.