  Stokes' manifolds, also known as wild character varieties, carry a natural
symplectic structure. Our goal is to provide explicit log-canonical coordinates
for these natural Poisson structures on the Stokes' manifolds of polynomial
connections of rank $2$, thus including the second Painlev\'e\ hierarchy. This
construction provides the explicit linearization of the Poisson structure first
discovered by Flaschka and Newell and then rediscovered and generalized by
Boalch. We show that, for a connection of degree $K$, the Stokes' manifold is a
cluster manifold of type $A_{2K}$. The main idea is then applied to express
explicitly also the log--canonical coordinates for the Poisson bracket
introduced by Ugaglia in the context of Frobenius manifolds and then also
applied by Bondal in the study of the symplectic groupoid of quadratic forms.
