Phase portraits of a family of Kolmogorov systems depending on six parameters

The paper proves that, under hypothesis (H2), the six-parameter Kolmogorov family ẋ = x(a0 − μ(c1 x + c2 z^2 + c3 z)), ż = z(c0 + c1 x + c2 z^2 + c3 z) has exactly 102 topologically distinct phase portraits on the Poincaré disc, after reducing by symmetries and excluding degenerate cases; see Theorem 1.1 and the definition of (H2) including μ ≠ −1, a0 c1 μ ≠ 0, and a0 + c0 μ ≠ 0 (the Darboux-invariant condition) . The paper develops the local analysis at finite singularities (with invariant axes and triangular Jacobians) and at infinity via Poincaré compactification and directional blow-ups, obtaining 47 local types at O1 and a simpler classification at O2, then glues them using separatrix-configuration equivalence, index constraints, and a precise contact-point result (Theorem 4.8) to enumerate G1–G102 . The candidate solution mirrors this approach almost step-by-step (reductions to H2 via reflections and time reversal; finite and infinite analyses; use of Theorem 2.1 on separatrix configurations; contact-point constraints; and index balance), differing only in minor phrasing (it says “at most one” horizontal contact when c1 ≠ 0, whereas the paper proves there is exactly one) .