  We establish a Liouville-type theorem for the elliptic and incompressible
Magnetic-B\'enard system defined over the entire three-dimensional space.
Specifically, we demonstrate the uniqueness of trivial solutions under the
condition that they belong to certain local Morrey spaces. Our results
generalize in two key directions: firstly, the Magnetic-B\'enard system
encompasses other significant coupled systems for which the Liouville problem
has not been previously studied, including the Boussinesq system, the
MHD-Boussinesq system, and the B\'enard system. Secondly, by employing local
Morrey spaces, our theorem applies to Lebesgue spaces, Lorentz spaces, Morrey
spaces, and certain weighted-Lebesgue spaces.
