In this paper, we consider an analytic cocycle $(\alpha,A)$ on $\T^d\times U(n)$. We prove that if the cocycle is conjugated to a constant cocycle $(\alpha,C)$ by a measurable conjugacy $(0,B)$, it is analytic reducible for almost all $ \alpha$ and $C$ provided $A$ is close to constant enough. Moreover if $B$ is continuous it is actually analytic.
15 pages