  In this work we study analytic Levi-flat hypersurfaces in complex algebraic
surfaces. First, we show that if this foliation admits chaotic dynamics (i.e.
if it does not admit a transverse invariant measure), then the connected
components of the complement of the hypersurface are modifications of Stein
domains. This allows us to extend the CR foliation to a singular algebraic
foliation on the ambient complex surface. We apply this result to prove, by
contradiction, that analytic Levi-flat hypersurfaces admitting a transverse
affine structure in a complex algebraic surface have a transverse invariant
measure. This leads us to conjecture that Levi-flat hypersurfaces in complex
algebraic surfaces that are diffeomorphic to a hyperbolic torus bundle over the
circle are fibrations by algebraic curves.
