On the correspondence between symmetries of two-dimensional autonomous dynamical systems and their phase plane realisations

The candidate’s steps (A–D) reproduce the paper’s three theorems and the lifting-PDE solution essentially verbatim. Step A matches Theorem 1 by computing Df, showing f∗(X(1)) = (f∗X)^{(1)}, with the ∂_{v′}-coefficient Duηv − v′Duηu agreeing with Eqs. (3.5)–(3.7) in the paper . Step B matches Theorem 2, deriving that f∗X is a symmetry of v′ = Ω via the determining equations and the quotient rule, consistent with Eq. (3.10) . Step C states the lifting condition Dtξ = (1/ωu)[(ωu∂u + ωv∂v)ζu − (ζu∂u + ζv∂v)ωu] and its equivalence with the ωv-form when ωv ≠ 0, as in Theorem 3 (Eqs. (4.2), (4.4)–(4.8)) . Step D solves the linear transport PDE ∂tξ + ωu∂uξ + ωv∂vξ = G(u,v) along characteristics, yielding ξ = ∫G ds + F with F constant on characteristics, exactly as Eqs. (4.9)–(4.12) . Minor omissions (e.g., the special ωv = 0 case treated explicitly in the paper) do not affect correctness.