  A violation of Local Lorentz Invariance (VLI) and hence the special theory of
relativity or a violation of equivalence principle (VEP) in the Kaon system
can, in principle, induce oscillations between $K^0$ and $\bar{K}^0$. We
construct a general formulation in which simultaneous pairwise diagonalization
of mass, momemtum, weak or gravitational eigenstates is not assumed. %and the
maximum attainable %velocities of the velocity eigenstates are different. We
discuss this problem in a general way and point out that, as expected, the VEP
and VLI contributions are indistinguishable. We then insist on the fact that
VEP or VLI can occur even when CPT is conserved. A possible CP violation of the
superweak type induced by VEP or VLI is introduced and discussed. We show that
the general VEP mechanism (or the VLI mechanism, but not both simultaneously),
with or without conserved CPT, could be clearly tested experimentally through
the energy dependence of the $K_L-K_S$ mass difference and of $\eta_{+-}$,
$\eta_{00}$, $\delta$. Constraints imposed by present experiments are
calculated.
