In this paper we consider the radiation properties of a pair of semi-infinite, parallel-plate ducts in which the inner duct is buried inside the outer duct. A Robin condition is applied to one of the inner walls (to represent an acoustic lining), while Neumann conditions are applied on all other surfaces. This leads to a matrix Wiener-Hopf problem, which requires the factorisation of a 3 × 3 matrix of a form which, to our knowledge, has not previously been considered in the literature and which is not directly amenable to standard pole-removal techniques. We derive the exact factorisation of this matrix here, and present results for the far-field scattered sound which show the effect of varying the properties of the wall lining.