Let \((X,C)\) be a pair where \(X\) is a germ of a normal two-dimensional singularity and \(C \subset X\) is a germ of smooth curve. Jaffe gave a complete description of such pairs up to isomorphisms when \(X\) is a rational double point in terms of the associated Dynkin diagram. In this paper that result is extended to a more general class of rational singularities. The description of the set of isomorphism classes of \((X,C)\) is in terms of the points that the fundamental cycle \(E\) in the minimal resolution of the singularity cuts out on the strict transform of \(C\) (via the resolution map). The result relies on some properties pointed out in a thorough study of the minimal resolution of the singularity.