  The problem of the existence of an analytic normal form near an equilibrium
point of an area-preserving map and analyticity of the associated coordinate
change is a classical problem in dynamical systems going back to Poincar\'e and
Siegel. One important class of examples of area-preserving maps consists of the
collision maps for planar billiards. Recently, Treschev discovered a formal
$\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetric billiard with locally linearizable
dynamics and conjectured its convergence. Since then, a Gevrey regularity for
such a billiard was proven by Wang and Zhang, but the original problem about
analyticity still remains open.
  We extend the class of billiards by relaxing the symmetry condition and
allowing conjugacies to non-linear analytic integrable normal forms. To keep
the formal solution unique, odd table derivatives and the normal form are
treated as parameters of the problem. We show that for the new problem, the
series of the billiard table diverge for general parameters by proving the
optimality of Gevrey bounds. The general parameter set is prevalent (in a
certain sense has full measure) and it contains an open set.
  Instead of considering the problem in a functional sense and iterating
approximation procedures, we employ formal power series methods and one-by-one
directly reconstruct all the Taylor coefficients of the table. In order to
prove that on an open set Taylor series diverges we define a Taylor recurrence
operator and prove that it has a cone property. All solutions in that cone are
only Gevrey regular and not analytic.
