The degenerate scale issue for 2D-boundary integral equations and boundary element methods has been already investigated for many cases when the properties of the medium are homogeneous. We address here the problem of several subdomains with different properties. Then, the domain decomposition into homogeneous subdomains gives rise to a system of BIEs. If there are n subdomains, (n bounded subdomains in the case of an interior problem, and bounded subdomains and one unbounded subdomain in the case of an exterior problem) there are n degenerate scales in both cases. For an interior problem, the n degenerate scales are the degenerate scales of the n subdomains. For an exterior problem, there are degenerate scales equal to the degenerate scales of the bounded subdomains and one intrinsic degenerate scale linked to the solution of a specific boundary value problem. Some properties of this intrinsic degenerate scale are investigated by analytical and numerical methods. The case of a half-plane is also studied according to the boundary condition along the line bounding the half-plane.