On the straightness of eigenvalue interactions

Solutions to eigenvalue problems come in two parts, an eigenvalue and an eigenvector, and these solution pairs occur at discrete points in the range of possible eigenvalues. Multiparameter eigenvalue problems similarly have solutions that have a dimension smaller by 1 than the space of the eigenvalues - solutions to a 2-parameter problem are discrete curves in a plane, and in general, solutions to an n-parameter problem are hypersurfaces in an n dimensional space. These curves/surfaces/hypersurfaces are eigenvalue interaction curves (/surfaces, etc.), and they might be flat. An unchanging eigenvector leads to a flat interaction, almost trivially. This paper addresses the question if an interaction is flat (in particular, if an interaction curve is straight), what conditions does this place on the eigenvector?