An approach is proposed in which the form of the partial waves in the decomposition of the displacement field is chosen in such a way that boundary conditions on the outer surfaces of a layered structure are matched exactly. So found basis functions are further used to solve for the remaining boundary conditions. A free bilayered plate composed of isotropic materials is considered as an example demonstrating the efficiency of the approach. Explicit analytic expressions describing the complete vibration spectrum and normalized amplitudes in the long wavelength regime are derived and their behavior in the full space of material parameters is discussed. Thus, it is found that properties of the fundamental flexural mode change in a strongly non-monotonous way. In particular, by increasing the thickness of added layer one may observe the same propagation velocity for up to three different values of the thickness ratio. It is shown that there exist an important topological property of the amplitudes which allow to establish a clear connection to the well known symmetries of Lamb's solutions for a single layer plate. Branches of the spectrum are classified accordingly and their evolution from long to short wavelength is discussed.