The paper considers transformation of the equation of motion in stresses for an incompressible Newtonian fluid. The aim of the transformation is to obtain more detailed equations that account for the impact of vortex (rotational) and linear (forward) flows on the process of viscous friction. The transformation method is based on adding zero to the expressions for shear stresses with subsequent distinguishing of rotor velocity function and derivatives characterizing the linear flow. This approach as a form of recording the original equation does not require any additional restrictions. The transformation has resulted in new systems of equations for viscous vortex and vortex-free flows as well as three-dimensional vortex. The obtained equations allow obtaining the known exact solution for the laminar flow (Poiseuille’s formula) and Euler’s differential equation for an ideal fluid. We have shown that the Navier-Stokes equation is a separate case of a more general equation for Newtonian fluid motion. The obtained equations and connections between them allow improvement of the mathematical description of the incompressible fluid flow.