  In this article we study deformations of a holomorphic foliation with a
generic non-rational first integral in the complex plane. We consider two
vanishing cycles in a regular fiber of the first integral with a non-zero self
intersection and with vanishing paths which intersect each other only at their
start points. It is proved that if the deformed holonomies of such vanishing
cycles commute then the deformed foliation has also a first integral. Our
result generalizes a similar result of Ilyashenko on the rigidity of
holomorphic foliations with a persistent center singularity. The main tools of
the proof are Picard-Lefschetz theory and the theory of iterated integrals for
such deformations.
