In this paper, we study general solutions to the Stokes equations in hydrodynamics and equations of equilibrium in the elasticity theory, obtain new forms for the representation of general solutions, and construct a numerical method for calculating the Stokes flows of a viscous fluid. By the representation of the solutions to the Stokes equations we mean the expression of these solutions in terms of harmonic functions. The new formulas include the representation in terms of the first order operator and a class of 27 representations. The symmetries of fluid flows with low rates and of elastic media equilibria have been found. The suggested numerical method involves the above-mentioned representations and a boundary integral equation of a particular type. It is essentially different from the known methods of boundary integral equations based on multipole solutions to the Stokes equations [1‐3]. The manner in which the solutions to the Stokes equations and to the equations of the elasticity theory are related to the solutions to the Laplace equations is considered in detail in [4] for the case of plane problems.