A waveguide occupies a domain G in ℝ³ with several cylindrical outlets to infinity; the boundary ∂G is assumed to be smooth. The dielectric ε and magnetic μ permittivities are matrix-valued functions smooth and positive definite in G. At every cylindrical outlet, the matrices e and μ tend, at infinity, to limit matrices independent of the axial variable. The limit matrices can be arbitrary smooth and positive definite matrix-valued functions of the transverse coordinates in the corresponding cylinder. In such a waveguide, the stationary Maxwell system with perfectly conducting boundary conditions and a real spectral parameter is considered. In the presence of charges and currents, the corresponding boundary value problem with radiation conditions turns out to be well posed. A unitary scattering matrix is also defined. The Maxwell system is extended to an elliptic system. The results for the Maxwell system are derived from those obtained for the elliptic problem.