Consider the scattering of an electromagnetic time-harmonic plane wave by an infinite cylinder having an open crack Γ and a bounded domain D in R 2 as cross section. We assume that the crack Γ is divided into two parts, and one of the two parts is (possibly) coated on one side by a material with surface impedance λ . Different boundary conditions are given on Γ and ∂ D . Applying potential theory, the problem can be reformulated as a boundary integral system. We obtain the existence and uniqueness of a solution to the system by using Fredholm theory.