A general representation for equations integrable by means of the inverse scattering transform is provided for by the Lax-pair operator equation, \[ \frac{\partial L}{\partial t}= LA - AL, \] where \(L\) and \(A\) are noncommuting operators whose coefficients depend on unknown functions governed by the integrable equations, and \(t\) is the evolution variable (time). In this paper, a generalized form of the operator evolution equation \(\frac{\partial L}{\partial t}= \Phi (L,t) +LA - AL, \) is considered, \(\Phi(L,t)\) is a polynomial with time-dependent coefficients (the same equation in an autonomous form was known earlier), and \(A=A(L(t),t)\) is a skew-symmetric matrix. Unlike the traditional Lax-pair equation, which gives rise to isospectral evolution of the operator \(L\), the evolution described by the modified equation is nonisospectral. This equation has applications to some integrable systems, for instance, an inhomogeneous version of the Toda lattice, a modified Lotka-Volterra equation, etc. In the paper, a solution to the Cauchy problem for the modified operator equation is constructed. In addition, this nonlinear equation is linearized by means of properly normalized moments of the spectral measure corresponding to the Jacobi matrix \(A\), and the uniqueness of the solution to the Cauchy problem is rigorously proven.