  Related to the classification of regular foliations in a complex algebraic
surface, we address the problem of classifying the complex surfaces which admit
a flat pencil of foliations. On this matter, a classification of flat pencils
which admit foliations with a first integral of genus one and isolated
singularities was done by Lins Neto. In this work, we complement Lins Neto's
work, by obtaining the classification of compact complex surfaces which have a
pencil with an invariant tangency set.
