  We discuss the hyperfine shifts of the Positronium levels in a relativistic
framework, starting from a two fermion wave equation where, in addition to the
Coulomb potential, the magnetic interaction between spins is described by a
Breit term. We write the system of four first order differential equations
describing this model. We discuss its mathematical features, mainly in relation
to possible singularities that may appear at finite values of the radial
coordinate. We solve the boundary value problems both in the singular and non
singular cases and we develop a perturbation scheme, well suited for numerical
computations, that allows to calculate the hyperfine shifts for any level,
according to well established physical arguments that the Breit term must be
treated at the first perturbative order. We discuss our results, comparing them
with the corresponding values obtained from semi-classical expansions.
