Explicitly solvable systems of two autonomous first-order Ordinary Differential Equations with homogeneous quadratic right-hand sides

The paper proves that any 2×2 autonomous ODE with homogeneous quadratic right-hand sides can be explicitly solved once it is linearly reduced to the Riccati cascade ẏ1 = y1^2, ẏ2 = ρ1 y1^2 + ρ2 y1 y2 + y2^2, and it provides the exact coefficient identities linking (A,B,ρ1,ρ2) to the original coefficients c_{nℓ} (its Proposition 2-2; formulas (4)–(6) show the transformation) . It then inverts those identities and derives two algebraic constraints (36a,b) on c_{nℓ} that are necessary for that reduction and, in effect, sufficient for computing (A,B,ρ1,ρ2) by solving a cubic for b21 (equations (34)–(41)) . A secondary simplification (that the cubic coefficient C3 vanishes under (36)) is observed numerically but not proven; the authors note this explicitly and emphasize that Cardano’s formulas already yield b21 anyway . The candidate model reproduces the forward identities (i) by direct expansion, offers a clean constructive sufficiency proof for (36) that bypasses the unproven C3=0 claim, and then solves the cascade exactly as in Proposition 2-1 . Hence both are correct; the paper’s path uses an algebraic inversion culminating in a (solvable) cubic, while the model gives a compatible, more geometric existence construction.