The paper reports an algorithm to analytically solve one of the problems in the mechanics of elastic bodies, which is associated with studying the natural vibrations of a composite two-stage plate whose concave part is smoothly aligned with the part of a constant thickness. We have defined patterns for stating the boundary and transitional conditions, which should be taken into account when considering the natural vibrations of a two-stage plate.The ratios have been obtained, which make it possible to study the distribution of deflections and determine the values of amplitudes of the curved vibrations of the plate. It was noted that the modes of vibrations are based on the symmetry and factorization methods that we had developed and refined earlier. Specifically, it has been found that the deflections can be explored through expressions that are derived through the sum of relevant solutions to two linear second-order differential equations with variable coefficients.Based on the proposed approach, a system consisting of eight homogeneous algebraic equations has been defined, which allowed us to build a frequency equation for the plate rigidly fixed along the inner contour and free along the outer contour. We have determined the values for the plate's natural frequencies for the first three modes of natural vibrations. Moreover, in order to verify and expand a set of plates of different configurations, the plates with two types of concave in their variable part have been considered. The new approaches and the ratios based on them could be useful for the further advancement of methods for solving similar problems in mathematical physics on natural values. A practical implementation is the problems about the vibrations of plates with variable thickness and of different modes