Noncommutative coboundary equations over integrable systems

We prove an analog of the Livshits theorem for real-analytic families of cocycles over an integrable system with values in a Banach algebra$ {{\mathscr G}} $or a Lie group. Namely, we consider an integrable dynamical system$ f:{{\mathscr M}} \equiv{{\mathbb T}}^d \times [-1, 1]^d\to {{\mathscr M}} $,$ f(\theta, I) = (\theta + I, I) $, and a real-analytic family of cocycles$ \eta_{\varepsilon} : {{\mathscr M}} \to {{\mathscr G}} $indexed by a complex parameter$ {\varepsilon} $in an open ball$ {{\mathscr E}}_\rho \subset {{\mathbb C}} $. We show that if$ \eta_{\varepsilon} $is close to identity and has trivial periodic data, i.e.,for each periodic point$ p = f^n p $and each$ {\varepsilon} \in {{\mathscr E}}_{\rho} $, then there exists a real-analytic family of maps$ \phi_{\varepsilon}: {{\mathscr M}} \to {{\mathscr G}} $satisfying the coboundary equation$   \eta_{\varepsilon}(\theta, I) = (\phi_{\varepsilon}\circ f(\theta, I))^{-1} \cdot \phi_{\varepsilon} (\theta, I)    $for all$ (\theta, I)\in {{\mathscr M}} $and$ {\varepsilon} \in {{\mathscr E}}_{\rho/2} $.We also show that if the coboundary equation above with an analytic left-hand side$ \eta_{\varepsilon} $has a solution in the sense of formal power series in$ {\varepsilon} $, then it has an analytic solution.