Abstract <p>An exact solution of the Navier–Stokes equations is constructed to describe unsteady shear isobaric flows of a viscous incompressible fluid. The original system of hydrodynamic equations is reduced to an overdetermined nonlinear system of partial differential equations. For this system, a nontrivial exact solution with functional arbitrariness is constructed in the Lin–Sidorov–Aristov class. For an analytical study of the properties of the system of equations, a generating solution was used, which allows replicating (multiplying) exact solutions for the equations of hydrodynamics. The description of unsteady flows is based on a modification of the Fourier variable separation method. Equations are derived to determine the structure of the hydrodynamic field. The development of the idea of constructing new types of exact solutions for a velocity field with a nonlinear dependence on two spatial coordinates is presented. The exact solutions announced in the paper allow describing flows of a vertically swirled fluid without preliminary swirling.</p>