To study the accuracy and quality of two-dimensional equations of refined theories for three-layer plates and shells, we obtain analytical solutions for the problem of eigenvibrations of a three-layer plate of infinite width. The plate is of symmetrical structure in the transversal direction. We employ the equations of theory of elasticity for describing the behavior of the core. All possible modes of vibrations may be subdivided into four categories. We derive transcendental equations for eigenfrequencies. Approximate values of roots are determined for all modes under consideration. Once the roots are known, we have an opportunity to use explicit analytical relationships and obtain the frequencies of vibrations with a desired accuracy. The results obtained are compared with known solutions. We make a conclusion that the model under analysis ensures high degree of accuracy in dynamical problems of mechanics of three-layer structures.