On a generalization of the It\^{o}-Wentzel formula for system of generalized It\^{o}'s SDEs and the stochastic first integral

Generalization of the It^{o}-Wentzel formula for the generalized It^{o}'s SDEs (It^{o}'s GSDEs) system with not centered measure is constructed. This construction is based on the basis of the stochastic kernel of integral transformation. The It^{o}'s GSDEs system for the kernel of the integral invariant is constructed. The concept of a stochastic first integral of the It^{o}'s GSDEs system with not centered measure is introduced. The conditions for the random function, that it’s the first integral of the set It^{o}'s GSDEs system, are defined.