The article considers non-one-dimensional convective layered flows of a viscous incompressible fluid with a spatial acceleration. The simulation is based on the equations of thermal convection in the Boussinesq approximation. The solution to these equations is sought in a generalized class of exact solutions in which all components of the velocity vector, pressure and temperature are presented in the form of complete linear forms along two Cartesian coordinates with non-linear (relative to the third Cartesian coordinate) coefficients. It is shown that for layered flows the system of defining relations reduces to an overdetermined system of ordinary differential equations. Two theorems, that justify the existence (under a special algebraic condition) and the uniqueness of the solution of the resulting overdetermined system, are formulated and proved.