The article considers thermal diffusion shear flows of a viscous incompressible fluid with spatial acceleration. The simulation uses a system of thermal diffusion equations (in the Boussinesq approximation), taking into account the Dufour effect. This system makes it possible to describe incompressible gases, for which this effect prevails, from a unified standpoint. It is shown that for shear flows, the system of equations under study is nonlinear and overdetermined. In view of the absence of a theorem on the existence and smoothness of the solution of the Navier–Stokes equation, the integration of the existing system seems to be an extremely difficult task. The article studies the question of the existence of a solution in the class of functions represented as complete linear forms in two Cartesian coordinates with non-linear (with respect to the third Cartesian coordinate) coefficients. It is shown that the system is non-trivially solvable under a certain condition (compatibility condition) constructed by the authors. The corresponding theorem is formulated and proven. These conclusions are illustrated by a comparison with the previously obtained results.