The generalized Itô–Venttsel' formula in the case of a noncentered Poisson measure, a stochastic first integral, and a first integral

We deduce an analog of the Itô–Venttsel' formula for an Itô system of generalized stochastic differential equations (GSDE) with noncentered measure on the basis of a stochastic kernel of an integral invariant. We construct a system of GSDE whose solution is a kernel of an integral invariant connected with a solution to GSDE with noncentered measure. We introduce the notion of a stochastic first integral of a system of GSDE with noncentered measure and find conditions under which a random function is a first integral of a given system of GSDE.