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The method of first integrals (MFI) based on the equation of motion for the displacement vector, or  based on the one for the traction vector was introduced  recently in order to find explicit secular equations of Rayleigh waves whose characteristic equations (i.e the equations determining the attenuation factor) are fully quartic or are of higher order (then the classical approach is not applicable). In this paper it is shown that, not only to Rayleigh waves,  the MFI can be applicable also to other waves by running it on the equations for mixed vectors. In particular: (i) By applying the MFI  to the equations for the displacement-traction vector we get the explicit dispersion equations of Stoneley waves in twinned crystals (ii)  Running the MFI on the equations for the traction-electric induction vector and the traction-electrical potential vector provides the explicit dispersion equations of SH-waves in piezoelastic materials. The obtained dispersion equations are identical with the ones previously derived using the method of polarization vector, but the procedure of driving them is more simple.

The problem of Rayleigh waves in compressible orthotropic elastic half-space overlaid by a thin elastic layer of which principal material axes are coincident have been researched by many scientists. However, the problem with the conditions that the half-space and the layer have only one common principal material axis that perpendicular to the layer while the remains are not identical has not gotten enough attention. This paper presents a traditional approach to obtain an approximate secular equation by approximately replacing the thin layer by effective boundary conditions of third-order. The wave then is considered as a Rayleigh wave propagating in an orthotropic half-space, without coating, subjected to the effective boundary conditions. This explicit approximate secular equation is potentially useful in non-damage assessment studies.