  Generalization of the It\^{o}-Wentzell formula for the generalized It\^{o}'s
SDE (It\^{o}'s GSDE) system with a non-centered measure is constructed on the
basis of the stochastic kernel of integral transformation. The It\^{o}'s GSDE
system for the kernel the solution of which is the kernel of the integral
invariant, is formed. This invariant is connected with the solution of the
It\^{o}'s GSDE system non-centered measure. The concept of a stochastic first
integral of the It\^{o}'s GSDE system with non-centered measure is introduced
and conditions that when being performed the random function is the first
integral of the set It\^{o}'s GSDE system are defined.
