Nonconmutative coboundary equations over integrable systems

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