$\mathcal{C}\mathcal{P}\mathcal{T}$-invariant two-fermion Dirac equation with extended hyperfine operator

For the S-states of muonium and positronium, the hyperfine shifts to the order ${\alpha }^6$ of a recently derived two-fermion equation with explicit $\cal CPT$-invariance are checked against the results of a nonrelativistic reduction, and the leading ${\alpha }^8$ shifts are calculated. An additional hyperfine operator is discovered which can milden the singularity for $r\to 0$ of the Dirac hyperfine operator, such that the resulting extended operator can be used nonperturbatively. The binding correction to magnetic moments is mentioned.