On a family of commutative Wick symbols

Conditions are found under which an infinite system of bosons with two-body interaction has a one-parameter family of integrals of the motion determined by the Wick symbol
$$ A(\lambda)=\exp\biggl[\dfrac1\lambda\mathbf P+\dfrac{h}{2\lambda^2}\mathbf V \biggr], $$
where $\mathbf V$ and $\mathbf P$ are the Wick symbols of the interaction and momentum operatore, respectively. Examples of interaction potentials for which these conditions are satisfied are given. The complete integrability of the corresponding classical systems is proved.