  The Veech group of a translation surface is the group of Jacobians of
orientation-preserving affine automorphisms of the surface. We present an
algorithm which constructs all translation surfaces with a given lattice Veech
group in any given stratum. In developing this algorithm, we give a new proof
of a finiteness result of Smillie and Weiss, namely that there are only
finitely many unit-area translation surfaces in any stratum with the same
lattice Veech group.
  Our methods can be applied to obtain obstructions of lattices being realized
as Veech groups in certain strata; in particular, we show that the square torus
is the only translation surface in any minimal stratum whose Veech group is all
of $\mathrm{SL}_2\mathbb{Z}$.
