We give sufficient conditions on the spectrum at the equilibrium point such that a Gevrey-$s$ family can be
Gevrey-$s$ conjugated to a simplified form, for $0\le s\le 1$. Local analytic results (i.e. $s=0$) are obtained
as a special case, including the classical Poincaré theorems and the analytic stable and unstable manifold
theorem. As another special case we show that certain center manifolds of analytic vector fields are of
Gevrey-$1$ type. We finally study the asymptotic properties of the conjugacy on a polysector with opening angles
smaller than $s\pi$ by considering a Borel-Laplace summation.