In 1891, Poincaré posed, in a series of three articles, a central question for the theory of differential equations: can one determine whether a polynomial differential equation in two complex variables is algebraically integrable? This problem, now known as the Poincaré Problem, has generated great interest and has been extensively studied ever since.

In this talk, we will begin with a brief historical review of the problem, presenting some important results and known counterexamples. The main purpose is to construct a new family of counterexamples, focusing on a particular class of foliations—those with a unique singular point. These examples are also relevant to other important topics, such as pencils on $\mathbb{C}\mathbb{P}^2$ with a unique base point; and fibrations on rational algebraic surfaces. This is joint work with Alexis Zamora from the Universidad de Zacatecas, Mexico.