Algaba, A.Fuentes, N.Gamero Gutiérrez, EstanislaoGarcía, C.2022-09-262022-09-262021Algaba, A., Fuentes, N., Gamero Gutiérrez, E. y García, C. (2021). On the integrability problem for the Hopf-zero singularity and its relation with the inverse Jacobi multiplier. Applied Mathematics and Computation, 405, Article number 126241.0096-3003https://hdl.handle.net/11441/137383Article number 126241In this paper we use the orbital normal form of the nondegenerate Hopf-zero singularity to obtain necessary conditions for the existence of first integrals for such singularity. Also, we analyze the relation between the existence of first integrals and of inverse Jacobi multipliers. Some algorithmic procedures for determining the existence of first integrals are presented, and they are applied to some families of vector fields.application/pdf13 p.engAttribution-NonCommercial-NoDerivatives 4.0 Internacionalhttp://creativecommons.org/licenses/by-nc-nd/4.0/Integrability problemNondegenerate Hopf-zero singularityQuasi-homogeneous normal formsAnalytic first integralsInverse Jacobi multipliersinfo:eu-repo/semantics/articleinfo:eu-repo/semantics/openAccesshttps://doi.org/10.1016/j.amc.2021.126241