  \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.
