We consider the problem on the diffraction of an electromagnetic wave at an interface of semiopen dielectric waveguides. We reduce this problem to various integral equations, in particular, to the regular integral equation with respect to the coefficients of expansion of the field by harmonics of continuous and discrete spectra. We use the method of an auxiliary overdetermined boundary problem. We establish the necessary and sufficient solvability conditions for overdetermined problems for the Helmholtz equation in a quadrant of a plane, in a half-strip, and in semi-infinite fiber domains. We formulate these conditions in terms of the dependencies between the boundary functions.