Explore integral conditions for the existence of first integrals in analytic deformations of complex saddle singularities in this 37-minute lecture from the Brazil-France School on Foliation Theory. Delivered by Víctor Arturo Martínez León from the Federal University of Latin American Integration, the talk is part of a collaborative event hosted simultaneously at CIRM and IMPA. Gain insights into foliation theory through this presentation, which is one of several mini-courses and short talks featuring Brazilian and French mathematicians. Access additional resources, including the event page and full video playlist, to further your understanding of this advanced mathematical topic.