  We consider an Einstein-Dirac-Maxwell system with two charged massless
spinors coupled with an electromagnetic field, and construct a family of exact
solutions to the system. The solution spacetime is an anisotropic
generalization of the static Einstein universe which has a global cosmic
magnetic field generated by the current of the spinors. The spacetime is
regarded as a toy model which describes global cosmic magnetic phenomena in the
universe. The spinors are induced from Sasakian quasi-Killing spinors, and the
total Dirac current flows along fibers of the Hopf-fibration. The magnetic
field is a contact magnetic field.
