Solvable systems of two coupled first-order ODEs with homogeneous cubic polynomial right-hand sides

The paper’s Proposition 1-1 solves the cubic planar system by an invertible linear change (y = a1 x1 + a2 x2, w = b1 x1 + b2 x2) that yields ẏ = y^3 and ẇ = w^3 + γ1 y w^2 + γ2 y^2 w + γ3 y^3, then sets u = w/y to separate and integrate, obtaining the implicit product law for u(t) together with the explicit y(t) = y(0)[1 − 2 y(0)^2 t]^{-1/2} (see formulas (6)–(12), (14)–(17) in the paper). The candidate solution reproduces the same steps: it verifies the linear identities implied by (6) to get ẏ = y^3 and the stated ẇ, reduces to u, performs the same partial-fraction integration using the λ-constraints, and reconstructs x via x = M^{-1}(y,w) (matching (8)–(10) and (16d), (17)). Minor caveats (c ≠ 0, y(0) ≠ 0 for u(0), and handling repeated roots of the cubic) are tacit in both. Overall, the argument and details coincide closely with the paper’s proof sketch of Proposition 1-1, and the solution is correct. Key loci in the paper: statement and formulas (6)–(10) and c ≠ 0 in (7e) , explicit y(t) and the product law for u(t) in (9), (10) , the u-ODE and verification in (16d), (17) , and λ-conditions (12a,b) .