We construct families of pencils in \(\mathbb {P}^2\) with a unique base point such that the associated foliations have a unique singular point. The families give counterexamples to the Poincaré Problem in the sense that for a given degree of the foliations, the degree of the pencils and their genus arbitrarily increases. The generic element of each family of pencils turns out to be isotrivial. We study other related topics, such as the instability of the pencils and the associated foliations. Finally, we prove the rationality of the variety defined by these families of pencils.
