It is shown that the existence of an analytic invariant in addition to the natural ones (momentum, energy and, in some cases, "number of particles") leads to the existence of infinitely many such invariants. Nevertheless, the existence of the additional motion invariant does not guarantee complete integrability. Complete integrability follows from the existence of an additional invariant only if the dispersion law is non-degenerative with respect to decays. If the dispersion law is degenerative, the "number of" motion invariants is insufficient for complete integrability and the S-matrix is factorized via decay processes "one into two" with real intermediate particles. In this paper we present also our results concerning enumeration of degenerative dispersion laws.