We construct a governing system of equations for three-layer plates with transversally soft core. The plates are of symmetrical structure through the thickness. The system of eight differential equations governs small free and natural oscillations in the case of large variability of stress and strain parameters. Using new set of unknown functions, we reduce this system of equations to the other system that can be used for studying, first, longitudinal modes of motion that are symmetrical (in-phase) with respect to the middle plane of the core, second, transversal modes of motion (transversal anti-phase modes), and third, any arbitrary modes of motion including in-phase bending modes. Based on the equations obtained, we derive a particular case of equations for free oscillations of a three-layer plate having no deformations and curvatures of outer layers. We present the formulas for three frequencies of such free oscillations for plates with an orthotropic core. One of these frequencies is the frequency of transversal anty-phase oscillations of outer layers due to deformation of transversal reduction of the core. This deformation is constant with the variation of space coordinates. Two other formulas can be used for determining the frequencies of the others anti-phase plane-parallel oscillations of the outer layers in the tangential directions. These frequencies are related to the shear deformations. We discuss the accuracy of the proposed mathematical model that is constructed for studying dynamic processes of deformation of three-layer plates and shells.