It is proved that if the spectrum and spectral measure of a semi-infinite Jacobi matrixL(t) change appropriately, thenL(t) satisfies a generalized Lax equation of the form, where Φ(λ, t) is a polynomial witht-dependent coefficients andA(L(t), t) is a skew-symmetric matrix which is determined by the evolution of the spectral data. Such an equation is equivalent to a wide class of generalized Toda lattices. The theory of Jacobi matrices gives rise to the procedure of solution of the corresponding Cauchy problem by the inverse spectral problem method. The linearization of this nonlinear equation in terms of the moments is established.