A polymer brush with chains consisting of anisotropic monomers, in a liquid consisting of free spherical particles, is studied by the molecular dynamics method. It is shown that, at some value of the concentration of spheres, a liquid-crystal or oriented-domain transition occurs in the brush. A densely-grafted brush and a sparsely-grafted brush are studied; for this system, the transition point seems not to depend strongly upon the grafting density. In the case of the densely-grafted brush, a liquid-crystal transition proceeds via an intermediate microphase segregated state. One microphase, located near the grafting surface, is characterized by high density and high degree of ordering of monomers. This part of the brush contains only a small concentration of spheres. On the periphery of the brush, a disordered microphase with low monomer density is located. This part of the brush is enriched with spherical particles. The two microphases are separated by a well-defined boundary. On increasing the sphere concentration, the boundary between microphases shifts towards the periphery, and eventually the ordered microphase extends through the whole brush volume. The monomers of the densely-grafted brush in their ordered state form different structures, namely, a smectic structure at relatively low values of sphere concentration, and a structure of orientationally ordered domains at the higher sphere densities.