In this paper, we classify the phase portraits in the Poincaré disc of all the Kolmogorov systems [Formula: see text] which depend on six parameters. We prove that these systems have [Formula: see text] topologically distinct phase portraits in the Poincaré disc. These systems are provided by a general three-dimensional Lotka–Volterra system with a rational first integral of degree two of the form [Formula: see text], restricted to each surface [Formula: see text] varying [Formula: see text], with the additional assumption that they have a Darboux invariant of the form [Formula: see text].

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