First integrals of the equilibrium equations in one-dimensional problems of nonlinear elasticity theory

The first integrals of the equilibrium equations in one-dimensional problems of nonlinear elasticity theory is discussed. The theory describes the bending, tension and torsion of an elastic anisotropic layer under the conservation laws of the finite strain theory. These integrals can be used to reduce the order of such systems of differential equations, obtaining exact solutions, and checking. The exact solution of the nonlinear problem of the pure bending of an elastic plate is universal, and valid for any material from the class of incompressible isotropic micropolar bodies.