  We identify conditions giving large natural classes of partial differential
operators for which it is possible to construct a complete set of Laplace
invariants. In order to do that we investigate general properties of
differential invariants of partial differential operators under gauge
transformations and introduce a sufficient condition for a set of invariants to
be complete. We also give a some mild conditions that guarantee the existence
of such a set. The proof is constructive. The method gives many examples of
invariants previously known in the literature as well as many new examples
including multidimensional.
