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We present an analytical calculation of the complete $\alpha^6$ correction toenergies of $nP$-levels of two-body systems consisting of the spin-$0$ or $1/2$extended-size particles with arbitrary masses and magnetic moments. Theobtained results apply to a wide class of two-body systems such as hydrogen,positronium, muonium, and pionic or aniprotonic helium ion. We found anadditional $\alpha^6$ correction for $nP$-levels of positronium, which waspreviously overlooked. Our results are also relevant for light muonic atoms,whose accurate theoretical predictions are required for extracting the nuclearcharge radii.

Two-body $P$-state energies at $\alpha^6$ order