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In this paper we use the orbital normal form of the nondegenerate Hopf-zerosingularity to obtain necessary conditions for the existence of first integralsfor such singularity. Also, we analyze the relation between the existence offirst integrals and of inverse Jacobi multipliers. Some algorithmic proceduresfor determining the existence of first integrals are presented, and they areapplied to some families of vector fields.

On the Integrability Problem for the Hopf-Zero singularity and its relation with the inverse Jacobi multiplier