We discuss the hyperfine shifts of the Positronium levels in a relativisticframework, starting from a two fermion wave equation where, in addition to theCoulomb potential, the magnetic interaction between spins is described by aBreit term. We write the system of four first order differential equationsdescribing this model. We discuss its mathematical features, mainly in relationto possible singularities that may appear at finite values of the radialcoordinate. We solve the boundary value problems both in the singular and nonsingular cases and we develop a perturbation scheme, well suited for numericalcomputations, that allows to calculate the hyperfine shifts for any level,according to well established physical arguments that the Breit term must betreated at the first perturbative order. We discuss our results, comparing themwith the corresponding values obtained from semi-classical expansions.
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