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Generalization of the It\^{o}-Wentzel formula for the generalized It\^{o}'sSDEs (It\^{o}'s GSDEs) system with not centered measure is constructed. Thisconstruction is based on the basis of the stochastic kernel of integraltransformation. The It\^{o}'s GSDEs system for the kernel of the integralinvariant is constructed. The concept of a stochastic first integral of theIt\^{o}'s GSDEs system with not centered measure is introduced. The conditionsfor the random function, that it's the first integral of the set It\^{o}'sGSDEs system, are defined.

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