Abstract <p>Generalized Cesaro formulas are found, allowing to determine the displacement field with an accuracy of up to quadratic polynomials through integro-differential operators from the strain tensor-deviator in 3D elasticity theory and 4D elasticity theory. It is shown that quadratures for the pseudovector (pseudotensor in 4D elasticity) of local rotations and deformation of volume change are determined by the strain deviator field with an accuracy of up to linear polynomials in coordinates. Conditions for the existence of the listed quadratures are presented in the form of five (nine for 4D) third-differential order compatibility equations with respect to the strain tensor-deviator components.</p>