We prove an analog of the Livshits theorem for real-analytic families of cocycles over an integrable system with values in a Banach algebra (Formula Presented) or a Lie group. Namely, we consider an integrable dynamical system (Formula Presented), and a real-analytic family of cocycles (Formula Presented) indexed by a complex parameter ε in an open ball (Formula Presented). We show that if ηε is close to identity and has trivial periodic data, i.e., (Formula Presented) for each periodic point p = fn p and each (Formula Presented), then there exists a real-analytic family of maps (Formula Presented) satisfying the coboundary equation (Formula Presented) for all (Formula Presented) and (Formula Presented). We also show that if the coboundary equation above with an analytic left-hand side ηε has a solution in the sense of formal power series in ε, then it has an analytic solution.