The monoenergetic transport equation with isotropic scattering and vacuum boundary conditions is applied to two-media spherical and plane systems. The mean-free-path is assumed to be the same even though the multiplication factors are different in both media. The two coupled integral equations that are obtained are numerically solved using the spatial Legendre polynomial method (Carlvik’s method). Tables of seven or more eigenvalues for various dimensions of the bodies are given, and the first five flux modes for some cases are plotted. In addition, for homogeneous systems, we present more accurate and higher eigenvalues than those thus far known.