Study of the convective flows of the fluids and gases fulfilled the infinite layers with fixed or free boundaries remains to be an actual problem. In the present paper convection of a fluid in an inclined layer with fixed, impermeable, moving boundaries is considered. The exact solutions of the differential Oberbeck-Boussinesq equations are constructed. These solutions generalize the solution of a problem of fluid convection in a horizontal layer [2]. The velocity profiles are presented. In the paper the following investigations are presented: interaction of gravitational convection and effects of the shear Couette flows; action of inclination angle on the flow behavior; flow features connected with internal heat sources.