A new family of exact solutions for the equations of magnetohydrodynamics of an incompressible fluid is presented. The convective fluid flows in a rectangular Cartesian coordinate system are considered. Convection in a conducting fluid is induced by thermal factors and a solute. Thus, the announced exact solution describes thermal diffusion in magnetic fluids. Exact solutions are constructed taking into account the cross dissipative effects of Soret and Dufour. In the article, the Lin-Sidorov-Aristov class is used to construct the exact solution. The velocity field and the magnetic field are described by linear forms with respect to two spatial coordinates. The coefficients of linear forms depend on the third coordinate and time. Pressure, temperature and solute concentration are described by quadratic forms. A system of equations for finding unknown functions for hydrodynamic fields is given. This system is overridden. The article presents an exact solution for determining unknown functions for describing steady Stokes flows of convection of a binary conducting fluid. When constructing an exact solution in a nonlinear system of magnetohydrodynamics, all the terms for the convective derivative were assumed to be equal to zero (there is no convective and diffusive mixing of a continuous medium).