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First integrals of a three-dimensional system in the case of one zero eigenvalue

In this paper we propose an efficient computational approach to find systems with a first integral within some families of polynomial systems of ordinary differential equations in the case when the matrix of the linear approximation has one zero eigenvalue while the other eigenvalues have negative real parts. We apply it to find conditions for the existence of first integrals for a three-dimensional system involving cubic polynomials.