  We describe a broadly applicable experimental proposal to search for the
violation of local Lorentz invariance (LLI) with atomic systems. The new scheme
uses dynamic decoupling and can be implemented in current atomic clocks
experiments, both with single ions and arrays of neutral atoms. Moreover, the
scheme can be performed on systems with no optical transitions, and therefore
it is also applicable to highly charged ions which exhibit particularly high
sensitivity to Lorentz invariance violation. We show the results of an
experiment measuring the expected signal of this proposal using a two-ion
crystal of $^{88}$Sr$^+$ ions. We also carry out a systematic study of the
sensitivity of highly charged ions to LLI to identify the best candidates for
the LLI tests.
