On a class of model systems that admit a lowering of powers in the Hamiltonian in the thermodynamic limit. I

Class of model systems is singled out, in which the interaction takes place only between the particles with some definite (and not all admissible) conservation laws momenta. It means analytically that the interaction Hamiltonian includes some additional $\delta$-functions, besides the usual $\delta$-function. It is proved that many-time correlation functions of model systems and systems with approximating Hamiltonian coincide in thermodynamical limit. The powers of polynomials in the interaction Hamiltonian of the approximating Hamiltonian are lower than the powers of polynomials in the interaction Hamiltonian of the model systems.