Need the masses of unstable particles and their antiparticles be equal in the CPT--invariant system?

We show that the real parts of diagonal matrix elements of the exact effective Hamiltonian governing the time evolution in the subspace of states of neutral kaons and similar particles can not be equal for $t > t_{0}$ ($t_{0}$ is the instant of creation of the pair $K_{0}, {\bar{K}}_{0}$) when the total system under consideration is CPT invariant but CP noninvariant. The unusual consequence of this result is that, contrary to the properties of stable particles, the masses of the unstable particle, e.g.. $K_{0}$, and its antiparticle, ${\bar{K}}_{0}$, need not be equal for $t \gg t_{0}$ in the case of preserved CPT and violated CP symmetries.