Creation of instability concept of evolution of the Earth and non-classical linearized approach to study this evolution; theoretical basis of mathematical-geodynamic model of development of internal structures of the Earth; mechanisms of formation of folding and fractures and their non-classical three-dimensional theory; exact solution of complete non-linear task of distribution of medium density in the Earth's interior and proof of instability of this solution for different rocks; solving of problem of sequence of realization of deformation process and phase transitions depending on density; Development of differential geomechanical criteria of distribution of elastic and physical parameters on depth of the internal structures of the Earth; evidence of violations of these criteria in the solid core of the Earth; for the purpose of interpreting the data of observations on selfoscillations of the Earth, the solution of problem of selfoscillations of the three-layered sphere, elastic parameters which are not uniform in depth; nonlinear theories of multi-componential and three-dimensional seismic waves; creation of non-classical theoretical base model of seismicity; instability concept of destruction of rocks of the Earth's interior; three-dimensional mathematical theory of loss of stability of equilibrium state in vicinities of wells, separate parts and internal structures of geological section; solution of nonlinear tasks on dynamics, mechanics of destruction, strength and stability of deformed solid bodies.