A family of exact solutions to the Navier-Stokes equations is constructed to describe nonuniform two-dimensional fluid motions. The superposition of the main unidirectional flow with the secondary flow is considered. The secondary flow is determined by suction or injection through permeable boundaries. This class of exact solutions is obtained by multiplicative and additive separation of variables. The flow of a viscous incompressible fluid is described by a polynomial of the horizontal (longitudinal) coordinate. The polynomial coefficients are functions of the vertical (transverse) coordinate and time. They are determined by a chain of homogeneous and inhomogeneous parabolic partial differential equations with a convective term. In the case of a steady flow, it is described by a system of ordinary differential equations with constant coefficients. An algorithm for integrating a system of ordinary differential equations for studying the steady motion of a viscous fluid is presented. In this case, all the functions defining the velocity are quasipolynomials since the system of ordinary differential equations has an Euler-form exact solution.