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We show that the real parts of diagonal matrix elements of the exacteffective Hamiltonian governing the time evolution in the subspace of states ofneutral 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 totalsystem under consideration is CPT invariant but CP noninvariant. The unusualconsequence of this result is that, contrary to the properties of stableparticles, the masses of the unstable particle, e.g.. $K_{0}$, and itsantiparticle, ${\bar{K}}_{0}$, need not be equal for $t \gg t_{0}$ in the caseof preserved CPT and violated CP symmetries.Comment: LaTex2e, New comments and references added, 21 page

NEED THE MASSES OF UNSTABLE PARTICLES AND THEIR ANTIPARTICLES BE EQUAL IN THE CPT-INVARIANT SYSTEM?