We attempt to construct the constitutive relations and the complete system of equations and boundary conditions and establish the general properties of a continuum on the basis of only the most general assumptions relative to the kinematic relationships in the examined continuum. As a result we find two quadratures of the Saint-Venant equations and determine their integrability conditions. We establish for the first time the third-order compatibility relations that are not further integrable in quadratures, written relative to the components of the strain deviator tensor. We construct the generalized Cesaro formulas and find the group of generalized displacements of the examined continuum, in which in addition to the solid-body translations and rotations there enter the global volume change and the volume-rotational displacements. The new group of variables is of interest from the viewpoint of the field theory of defects. We obtain by the variational method the constitutive relations and the complete system of resolving equations for linearly elastic media with the introduced constraints, without any additional assumptions. These relations and equations describe media with nonparity of the tangential stresses and with elastic constraints of the distributed elastic foundation type that are distributed through the volume and over the surface. It is shown that in the particular case with additional kinematic constraints the proposed media description algorithm makes it possible to construct the traditional variants of the moment theory of elasticity. We establish the system of linear and quadratic isoperimetric relations, the first of which are a generalization of the global continuum equilibrium equations corresponding to the new group of generalized variables in the Cesaro formula. The quadratic isoperimetric relations are by construction the conservation conditions. We examine as applications the problems of determining the effective characteristics of nonhomogeneous media on the basis of the quadratic isoperimetric relations. We construct the system of thermodynamic constitutive relations and the complete system of resolving incremental equations for the nonlinear coupled problem of the deformation of structural elements made of shape memory materials.